Patchworking real algebraic varieties

Oleg Viro Mathematics Department, Stony Brook University, NY 11794
oleg.viro@@gmail.com
1991 Mathematics Subject Classification:
14G30, 14H99; Secondary 14H20, 14N10

Introduction

This paper is a translation of the first chapter of my dissertation 11This is not a Ph D., but a dissertation for the degree of Doctor of Physico-Mathematical Sciences. In Russia there are two degrees in mathematics. The lower, degree corresponding approximately to Ph D., is called Candidate of Physico-Mathematical Sciences. The high degree dissertation is supposed to be devoted to a subject distinct from the subject of the Candidate dissertation. My Candidate dissertation was on interpretation of signature invariants of knots in terms of intersection form of branched covering spaces of the 4-ball. It was defended in 1974. which was defended in 1983. I do not take here an attempt of updating.

The results of the dissertation were obtained in 1978-80, announced in [Vir79a]; [Vir79b]; [Vir80], a short fragment was published in detail in [Vir83a] and a considerable part was published in paper [Vir83b]. The later publication appeared, however, in almost inaccessible edition and has not been translated into English.

In [Vir89] I presented almost all constructions of plane curves contained in the dissertation, but in a simplified version: without description of the main underlying patchwork construction of algebraic hypersurfaces. Now I regard the latter as the most important result of the dissertation with potential range of application much wider than topology of real algebraic varieties. It was the subject of the first chapter of the dissertation, and it is this chapter that is presented in this paper.

In the dissertation the patchwork construction was applied only in the case of plane curves. It is developed in considerably higher generality. This is motivated not only by a hope on future applications, but mainly internal logic of the subject. In particular, the proof of Main Patchwork Theorem in the two-dimensional situation is based on results related to the three-dimensional situation and analogous to the two-dimensional results which are involved into formulation of the two-dimensional Patchwork Theorem. Thus, it is natural to formulate and prove these results once for all dimensions, but then it is not natural to confine Patchwork Theorem itself to the two-dimensional case. The exposition becomes heavier because of high degree of generality. Therefore the main text is prefaced with a section with visualizable presentation of results. The other sections formally are not based on the first one and contain the most general formulations and complete proofs.

In the last section another, more elementary, approach is expounded. It gives more detailed information about the constructed manifolds, having not only topological but also metric character. There, in particular, Main Patchwork Theorem is proved once again.

I am grateful to Julia Viro who translated this text.

1. Patchworking plane real algebraic curves

This Section is introductory. I explain the character of results staying in the framework of plane curves. A real exposition begins in Section 2. It does not depend on Section 1. To a reader who is motivated enough and does not like informal texts without proofs, I would recommend to skip this Section.

1.1. The case of smallest patches

We start with the special case of the patchworking. In this case the patches are so simple that they do not demand a special care. It purifies the construction and makes it a straight bridge between combinatorial geometry and real algebraic geometry.

1.1.A  Initial Data.

Let m be a positive integer number [it is the degree of the curve under construction]. Let Δ be the triangle in R2 with vertices (0,0), (m,0), (0,m) [it is a would-be Newton diagram of the equation]. Let T be a triangulation of Δ whose vertices have integer coordinates. Let the vertices of T be equipped with signs; the sign (plus or minus) at the vertex with coordinates (i,j) is denoted by σi,j.

See Figure 1.

Figure 1.

For ε,δ=±1 denote the reflection R2R2:(x,y)(εx,δy) by Sε,δ. For a set AR2, denote Sε,δ(A) by Aε,δ (see Figure 2). Denote a quadrant {(x,y)R2|εx>0,δy>0} by Qε,δ.

Figure 2.

The following construction associates with Initial Data 1.1.A  above a piecewise linear curve in the projective plane.

1.1.B  Combinatorial patchworking.

Take the square Δ made of Δ and its mirror images Δ+, Δ+ and Δ. Extend the triangulation T of Δ to a triangulation T of Δ symmetric with respect to the coordinate axes. Extend the distribution of signs σi,j to a distribution of signs on the vertices of the extended triangulation which satisfies the following condition: σi,jσεi,δjεiδj=1 for any vertex (i,j) of T and ε,δ=±1. (In other words, passing from a vertex to its mirror image with respect to an axis we preserve its sign if the distance from the vertex to the axis is even, and change the sign if the distance is odd.)22More sophisticated description: the new distribution should satisfy the modular property: g(σi,jxiyj)=σg(i,j)xiyj for g=Sεδ (in other words, the sign at a vertex is the sign of the corresponding monomial in the quadrant containing the vertex).

If a triangle of the triangulation T has vertices of different signs, draw the midline separating the vertices of different signs. Denote by L the union of these midlines. It is a collection of polygonal lines contained in Δ. Glue by S the opposite sides of Δ. The resulting space Δ¯ is homeomorphic to the projective plane RP2. Denote by L¯ the image of L in Δ¯.

Figure 3. Combinatorial patchworking of the initial data shown in Figure 1

Let us introduce a supplementary assumption: the triangulation T of Δ is convex. It means that there exists a convex piecewise linear function ν:ΔR which is linear on each triangle of T and not linear on the union of any two triangles of T. A function ν with this property is said to convexify T.

In fact, to stay in the frameworks of algebraic geometry we need to accept an additional assumption: a function ν convexifying T should take integer value on each vertex of T. Such a function is said to convexify T over Z. However this additional restriction is easy to satisfy. A function ν:ΔR convexifying T is characterized by its values on vertices of T. It is easy to see that this provides a natural identification of the set of functions convexifying T with an open convex cone of RN where N is the number of vertices of T. Therefore if this set is not empty, then it contains a point with rational coordinates, and hence a point with integer coordinates.

1.1.C  Polynomial patchworking.

Given Initial Data m, Δ, T and σi,j as above and a function ν convexifying T over Z. Take the polynomial

b(x,y,t)=(i,j)\enspace runs %over\quad vertices of Tσi,jxiyjtν(i,j).

and consider it as a one-parameter family of polynomials: set bt(x,y)=b(x,y,t). Denote by Bt the corresponding homogeneous polynomials:

Bt(x0,x1,x2)=x0mbt(x1/x0,x2/x0).
1.1.D  Patchwork Theorem.

Let m, Δ, T and σi,j be an initial data as above and ν a function convexifying T over Z. Denote by bt and Bt the non-homogeneous and homogeneous polynomials obtained by the polynomial patchworking of these initial data and by L and L¯ the piecewise linear curves in the square Δ and its quotient space Δ¯ respectively obtained from the same initial data by the combinatorial patchworking.

Then there exists t0>0 such that for any t(0,t0] the equation bt(x,y)=0 defines in the plane R2 a curve ct such that the pair (R2,ct) is homeomorphic to the pair (Δ,L) and the equation Bt(x0,x1,x2)=0 defines in the real projective plane a curve Ct such that the pair (RP2,Ct) is homeomorphic to the pair (Δ¯,L¯).

 Example 1.1.E  .

Construction of a curve of degree 2 is shown in Figure 3. The broken line corresponds to an ellipse. More complicated examples with a curves of degree 6 are shown in Figures 4, 5.

Figure 4. Harnack’s curve of degree 6.

Figure 5. Gudkov’s curve of degree 6.

For more general version of the patchworking we have to prepare patches. Shortly speaking, the role of patches was played above by lines. The generalization below is a transition from lines to curves. Therefore we proceed with a preliminary study of curves.

1.2. Logarithmic asymptotes of a curve

As is known since Newton’s works (see [New67]), behavior of a curve {(x,y)R2|a(x,y)=0} near the coordinate axes and at infinity depends, as a rule, on the coefficients of a corresponding to the boundary points of its Newton polygon Δ(a). We need more specific formulations, but prior to that we have to introduce several notations and discuss some notions.

For a set ΓR2 and a polynomial a(x,y)=ωZ2aωxω1yω2, denote the polynomial ωΓZ2aωxω1yω2 by aΓ. It is called the Γ-truncation of a.

For a set UR2 and a real polynomial a in two variables, denote the curve {(x,y)U|a(x,y)=0} by VU(a).

The complement of the coordinate axes in R2, i.e. a set {(x,y)R2|xy0}, is denoted33This notation is motivated in Section 2.3 below. by RR2.

Denote by l the map RR2R2 defined by formula l(x, y)=(ln|x|,ln|y|). It is clear that the restriction of l to each quadrant is a diffeomorphism.

A polynomial in two variables is said to be quasi-homogeneous if its Newton polygon is a segment. The simplest real quasi-homogeneous polynomials are binomials of the form αxp+βyq where p and q are relatively prime. A curve VRR2(a), where a is a binomial, is called quasiline. The map l transforms quasilines to lines. In that way any line with rational slope can be obtained. The image l(VRR2(a)) of the quasiline VRR2(a) is orthogonal to the segment Δ(a).

It is clear that any real quasi-homogeneous polynomial in 2 variables is decomposable into a product of binomials of the type described above and trinomials without zeros in RR2. Thus if a is a real quasi-homogeneous polynomial then the curve VRR2(a) is decomposable into a union of several quasilines which are transformed by l to lines orthogonal to Δ(a).

A real polynomial a in two variables is said to be peripherally nondegenerate if for any side Γ of its Newton polygon the curve VRR2(aΓ) is nonsingular (it is a union of quasilines transformed by l to parallel lines, so the condition that it is nonsingular means absence of multiple components). Being peripherally nondegenerate is typical in the sense that among polynomials with the same Newton polygons the peripherally nondegenerate ones form nonempty set open in the Zarisky topology.

For a side Γ of a polygon Δ, denote by DCΔ(Γ) a ray consisting of vectors orthogonal to Γ and directed outside Δ with respect to Γ (see Figure 6 and Section 2.2).

Figure 6.

The assertion in the beginning of this Section about behavior of a curve nearby the coordinate axes and at infinity can be made now more precise in the following way.

1.2.A  .

Let ΔRR2 be a convex polygon with nonempty interior and sides Γ1, …, Γn. Let a be a peripherally nondegenerate real polynomial in 2 variables with Δ(a)=Δ. Then for any quadrant URR2 each line contained in l(VU(aΓi) with i=1,…,n is an asymptote of l(VU(a)), and l(VU(a)) goes to infinity only along these asymptotes in the directions defined by rays DCΔ(Γi).

Theorem generalizing this proposition is formulated in Section 6.3 and proved in Section 6.4. Here we restrict ourselves to the following elementary example illustrating 1.2.A  .

 Example 1.2.B  .

Consider the polynomial a(x,y)=8x3x2+4y2. Its Newton polygon is shown in Figure 6. In Figure 7 the curve VR2(a) is shown. In Figure 8 the rays DCΔ(Γi) and the images of VU(a) and VU(aΓi) under diffeomorphisms l|U:UR2 are shown, where U runs over the set of components of RR2 (i.e. quadrants). In Figure 9 the images of DCΔ(Γi) under l and the curves VR2(a) and VR2(aΓi) are shown.

Figure 7.

Figure 8.

Figure 9.

1.3. Charts of polynomials

The notion of a chart of a polynomial is fundamental for what follows. It is introduced naturally via the theory of toric varieties (see Section 3). Another definition, which is less natural and applicable to a narrower class of polynomials, but more elementary, can be extracted from the results generalizing Theorem 1.2.A  (see Section 6). In this Section, first, I try to give a rough idea about the definition related with toric varieties, and then I give the definitions related with Theorem 1.2.A  with all details.

To any convex closed polygon ΔR2 with vertices whose coordinates are integers, a real algebraic surface RΔ is associated. This surface is a completion of RR2 (=(R0)2). The complement RΔRR2 consists of lines corresponding to sides of Δ. From the topological viewpoint RΔ can be obtained from four copies of Δ by pairwise gluing of their sides. For a real polynomial a in two variables we denote the closure of VRR2(a) in RΔ by VRΔ(a). Let a be a real polynomial in two variables which is not quasi-homogeneous. (The latter assumption is not necessary, it is made for the sake of simplicity.) Cut the surface RΔ(a) along lines of RΔ(a)RR2 (i.e. replace each of these lines by two lines). The result is four copies of Δ(a) and a curve lying in them obtained from VRΔ(a)(a). The pair consisting of these four polygons and this curve is a chart of a.

Recall that for ε,δ=±1 we denote the reflection R2R2:(x,y)(εx,δy) by Sε,δ. For a set AR2 we denote Sε,δ(A) by Aε,δ (see Figure 2). Denote a quadrant {(x,y)R2|εx>0,δy>0} by Qε,δ.

Now define the charts for two classes of real polynomials separately.

First, consider the case of quasi-homogeneous polynomials. Let a be a quasi-homogeneous polynomial defining a nonsingular curve VRR2(a). Let (w1,w2) be a vector orthogonal to Δ=Δ(a) with integer relatively prime coordinates. It is clear that in this case VR2(a) is invariant under S(1)w1,(1)w2. A pair (Δ, υ) consisting of Δ and a finite set υΔ is called a chart of a, if the number of points of υΔε,δ is equal to the number of components of VQε,δ(a) and υ is invariant under S(1)w1,(1)w2 (remind that VR2(a) is invariant under the same reflection).

 Example 1.3.A  .

In Figure 10 it is shown a curve VR2(a) with a(x,y)=2x6yx4y22x2y3+y4=(x2y)(x2+y)(2x2y)y, and a chart of a.

Figure 10.

Now consider the case of peripherally nondegenerate polynomials with Newton polygons having nonempty interiors. Let Δ, Γ1,,Γn and a be as in 1.2.A  . Then, as it follows from 1.2.A  , there exist a disk DR2 with center at the origin and neighborhoods D1,,Dn of rays DCΔ(Γ1),,DCΔ(Γn) such that the curve VRR2(a) lies in l1(DD1Dn) and for i=1,,n the curve Vl1(DiD)(a) is approximated by Vl1(DiD)(aΓi) and can be contracted (in itself) to Vl1(DiD)(a).

A pair (Δ, υ) consisting of Δ and a curve υΔ is called a chart of a if

  1. (1)

    for i=1,,n the pair (Γi, Γiυ) is a chart of aΓi and

  2. (2)

    for ε, δ=±1 there exists a homeomorphism hε,δ:DΔ such that υΔε,δ=Sε,δhε,δl(Vl1(D)Qε,δ(a)) and hε,δ(DDi)Γi for i=1,,n.

It follows from 1.2.A  that any peripherally nondegenerate real polynomial a with IntΔ(a) has a chart. It is easy to see that the chart is unique up to a homeomorphism ΔΔ preserving the polygons Δε,δ, their sides and their vertices.

 Example 1.3.B  .

In Figure 11 it is shown a chart of 8x3x2+4y2 which was considered in 1.2.B  .

Figure 11.
1.3.C  Generalization of Example 1.3.B  .

Let

a(x,y)=a1xi1yj1+a2xi2yj2+a3xi3yj3

be a non-quasi-homogeneous real polynomial (i. e., a real trinomial whose the Newton polygon has nonempty interior). For ε,δ=±1 set

σεik,δjk=sign(akεikδjk).

Then the pair consisting of Δ and the midlines of Δε,δ separating the vertices (εik,δjk) with opposite signs σεik,δjk is a chart of a.

Proof.

Consider the restriction of a to the quadrant Qε,δ. If all signs σεik,δjk are the same, then aQε,δ is a sum of three monomials taking values of the same sign on Qε,δ. In this case VQε,δ(a) is empty. Otherwise, consider the side Γ of the triangle Δ on whose end points the signs coincide. Take a vector (w1,w2) orthogonal to Γ. Consider the curve defined by parametric equation t(x0tw1,y0tw2). It is easy to see that the ratio of the monomials corresponding to the end points of Γ does not change along this curve, and hence the sum of them is monotone. The ratio of each of these two monomials with the third one changes from 0 to monotonically. Therefore the trinomial divided by the monomial which does not sit on Γ changes from to 1 continuously and monotonically. Therefore it takes the zero value once. Curves t(x0tw1,y0tw2) are disjoint and fill Qε,δ. Therefore, the curve VQε,δ(a) is isotopic to the preimage under Sε,δhε,δl of the midline of the triangle Δε,δ separating the vertices with opposite signs.

1.3.D  .

If a is a peripherally nondegenerate real polynomial in two variables then the topology of a curve VRR2(a) (i.e. the topological type of pair (RR2,VRR2(a))) and the topology of its closure in R2, RP2 and other toric extensions of RR2 can be recovered from a chart of a.

The part of this proposition concerning to VRR2(a) follows from 1.2.A  . See below Sections 2 and 3 about toric extensions of RR2 and closures of VRR2(a) in them. In the next Subsection algorithms recovering the topology of closures of VRR2(a) in R2 and RP2 from a chart of a are described.

1.4. Recovering the topology of a curve from a chart of the polynomial

First, I shall describe an auxiliary algorithm which is a block of two main algorithms of this Section.

1.4.A  Algorithm. Adjoining a side with normal vector (α,β).

Initial data: a chart (Δ,υ) of a polynomial.

If Δ (=Δ++) has a side Γ with (α,β)DCΔ(Γ) then the algorithm does not change (Δ,υ). Otherwise:

1. Drawn the lines of support of Δ orthogonal to (α,β).

2. Take the point belonging to Δ on each of the two lines of support, and join these points with a segment.

3. Cut the polygon Δ along this segment.

4. Move the pieces obtained aside from each other by parallel translations defined by vectors whose difference is orthogonal to (α,β).

5. Fill the space obtained between the pieces with a parallelogram whose opposite sides are the edges of the cut.

6. Extend the operations applied above to Δ to Δ using symmetries Sε,δ.

7. Connect the points of edges of the cut obtained from points of υ with segments which are parallel to the other pairs of the sides of the parallelograms inserted, and adjoin these segments to what is obtained from υ. The result and the polygon obtained from Δ constitute the chart produced by the algorithm.

 Example 1.4.B  .

In Figure 12 the steps of Algorithm 1.4.A  are shown. It is applied to (α,β)=(1,0) and the chart of 8x3x2+4y2 shown in Figure 11.

Figure 12.

Application of Algorithm 1.4.A  to a chart of a polynomial a (in the case when it does change the chart) gives rise a chart of polynomial

(xβyα+xβyα)x|β|y|α|a(x,y).

If Δ is a segment (i.e. the initial polynomial is quasi-homogeneous) and this segment is not orthogonal to the vector (α,β) then Algorithm 1.4.A  gives rise to a chart consisting of four parallelograms, each of which contains as many parallel segments as components of the curve are contained in corresponding quadrant.

1.4.C  Algorithm.

Recovering the topology of an affine curve from a chart of the polynomial. Initial data: a chart (Δ,υ) of a polynomial.

1. Apply Algorithm 1.4.A  with (α,β)=(0,1) to (Δ,υ). Assign the former notation (α,β) to the result obtained.

2. Apply Algorithm 1.4.A  with (α,β)=(0,1) to (Δ,υ). Assign the former notation (α,β) to the result obtained.

3. Glue by S+, the sides of Δ+,δ, Δ,δ which are faced to each other and parallel to (0,1) (unless the sides coincide).

4. Glue by S,+ the sides of Δε,+, Δε, which are faced to each other and parallel to (1,0) (unless the sides coincide).

5. Contract to a point all sides obtained from the sides of Δ whose normals are directed into quadrant P,.

6. Remove the sides which are not touched on in blocks 3, 4 and 5.

Algorithm 1.4.C  turns the polygon Δ to a space Δ which is homeomorphic to R2, and the set υ to a set υΔ such that the pair (Δ,υ) is homeomorphic to (R2,ClVRR2(a)), where Cl denotes closure and a is a polynomial whose chart is (Δ,υ).

Figure 13.
 Example 1.4.D  .

In Figure 13 the steps of Algorithm 1.4.C  applying to a chart of polynomial 8x3yx2y+4y3 are shown.

1.4.E  Algorithm.

Recovering the topology of a projective curve from a chart of the polynomial. Initial data: a chart (Δ,υ) of a polynomial.

1. Block 1 of Algorithm 1.4.C  .

2. Block 2 of Algorithm 1.4.C  .

3. Apply Algorithm 1.4.A  with (α,β)=(1,1) to (Δ,υ). Assign the former notation (Δ,υ) to the result obtained.

4. Block 3 of Algorithm 1.4.C  .

5. Block 4 of Algorithm 1.4.C  .

6. Glue by S, the sides of Δ++ and Δ which are faced to each other and orthogonal to (1,1).

7. Glue by S, the sides of Δ+ and Δ+ which are faced to each other and orthogonal to (1,1).

8. Block 5 of Algorithm 1.4.C  .

9. Contract to a point all sides obtained from the sides of Δ with normals directed into the angle {(x,y)R2|x<0,y+x>0}.

10. Contract to a point all sides obtained from the sides of Δ with normals directed into the angle {(x, y)R2|y<0,y+x>0}.

Algorithm 1.4.E  turns polygon Δ to a space Δ which is homeomorphic to projective plane RP2, and the set υ to a set υ such that the pair (Δ,υ) is homeomorphic to (RP2,VRR2(a)), where a is the polynomial whose chart is the initial pair (Δ,υ).

1.5. Patchworking charts

Let a1,,as be peripherally nondegenerate real polynomials in two variables with IntΔ(ai)IntΔ(aj)= for ij. A pair (Δ,υ) is said to be obtained by patchworking if Δ=i=1sΔ(ai) and there exist charts (Δ(ai),υi) of a1,,as such that υ=i=1sυi.

 Example 1.5.A  .

In Figure 11 and 1.5 charts of polynomials 8x3x2+4y2 and 4y2x2+1 are shown. In Figure 1.5 the result of patchworking these charts is shown.

\epsfboxpwf11.eps\epsfboxpwf12.eps\sc Figure ???\sc Figure ???

1.6. Patchworking polynomials

Let a1,,as be real polynomials in two variables with IntΔ(ai)IntΔ(aj)= for ij and aiΔ(ai)Δ(aj)=ajΔ(ai)Δ(aj) for any i, j. Suppose the set Δ=i=1sΔ(ai) is convex. Then, obviously, there exists the unique polynomial a with Δ(a)=Δ and aΔ(ai)=ai for i=1,,s.

Let ν:ΔR be a convex function such that:

  1. (1)

    restrictions ν|Δ(ai) are linear;

  2. (2)

    if the restriction of ν to an open set is linear then the set is contained in one of Δ(ai);

  3. (3)

    ν(ΔZ2)Z.

Then ν is said to convexify the partition Δ(a1),,Δ(as) of Δ.

If a(x,y)=ωZ2aωxω1yω2 then we put

bt(x,y)=ωZ2aωxω1yω2tν(ω1,ω2)

and say that polynomials bt are obtained by patchworking a1,,as by ν.

 Example 1.6.A  .

Let a1(x,y)=8x3x2+4y2, a2(x,y)=4y2x2+1 and

ν(ω1,ω2)={0,if ω1+ω222ω1ω2,if ω1+ω22.

Then bt(x,y)=8x3x2+4y2+t2.

1.7. The Main Patchwork Theorem

A real polynomial a in two variables is said to be completely nondegenerate if it is peripherally nondegenerate (i.e. for any side Γ of its Newton polygon the curve VRR2(aΓ) is nonsingular) and the curve VRR2(a) is nonsingular.

1.7.A  .

If a1,,as are completely nondegenerate polynomials satisfying all conditions of Section 1.6, and bt are obtained from them by patchworking by some nonnegative convex function ν convexifying Δ(a1),,Δ(as), then there exists t0>0 such that for any t(0,t0] the polynomial bt is completely nondegenerate and its chart is obtained by patchworking charts of a1,,as.

By 1.3.C  , Theorem 1.7.A  generalizes Theorem 1.1.D  . Theorem generalizing Theorem 1.7.A  is proven in Section 4.3. Here we restrict ourselves to several examples.

 Example 1.7.B  .

Polynomial 8x3x2+4y2+t2 with t>0 small enough has the chart shown in Figure 1.5. See examples 1.5.A  and 1.6.A  .

In the next Section there are a number of considerably more complicated examples demonstrating efficiency of Theorem 1.7.A  in the topology of real algebraic curves.

1.8. Construction of M-curves of degree 6

One of central points of the well known 16th Hilbert’s problem [Hil01] is the problem of isotopy classification of curves of degree 6 consisting of 11 components (by the Harnack inequality [Har76] the number of components of a curve of degree 6 is at most 11). Hilbert conjectured that there exist only two isotopy types of such curves. Namely, the types shown in Figure Figure 16 (a) and (b). His conjecture was disproved by Gudkov [GU69] in 1969. Gudkov constructed a curve of degree 6 with ovals’ disposition shown in Figure 16 (c) and completed solution of the problem of isotopy classification of nonsingular curves of degree 6. In particular, he proved, that any curve of degree 6 with 11 components is isotopic to one of the curves of Figure 16.

Figure 16.

Gudkov proposed twice — in [Gud73] and [Gud71] — simplified proofs of realizability of the third isotopy type. His constructions, however, are essentially more complicated than the construction described below, which is based on 1.7.A  and besides gives rise to realization of the other two types, and, after a small modification, realization of almost all isotopy types of nonsingular plane projective real algebraic curves of degree 6 (see [Vir89]).

Figure 17.

Construction In Figure 17 two curves of degree 6 are shown. Each of them has one singular point at which three nonsingular branches are second order tangent to each other (i.e. this singularity belongs to type J10 in the Arnold classification [AVGZ82]). The curves of Figure 17 (a) and (b) are easily constructed by the Hilbert method [Hil91], see in [Vir89], Section 4.2.

Choosing in the projective plane various affine coordinate systems, one obtains various polynomials defining these curves. In Figures 1.8 and 1.8 charts of four polynomials appeared in this way are shown. In 1.8 the results of patchworking charts of Figures 1.8 and 1.8 are shown. All constructions can be done in such a way that Theorem 1.7.A  (see [Vir89], Section 4.2) may be applied to the corresponding polynomials. It ensures existence of polynomials with charts shown in Figure 1.8.

\epsfboxpwf15.eps\sc Figure ???\epsfboxpwf16.eps\epsfboxpwf17.eps\sc Figure ???\sc Figure ???

1.9. Behavior of curve VRR2(bt) as t0

Let a1,,as, Δ and ν be as in Section 1.6. Suppose that polynomials a1,,as are completely nondegenerate and ν|Δ(a1)=0. According to Theorem 1.7.A  , the polynomial bt with sufficiently small t>0 has a chart obtained by patchworking charts of a1,,as. Obviously, b0=a1 since ν|Δ(a1)=0. Thus when t comes to zero the chart of a1 stays only, the other charts disappear.

How do the domains containing the pieces of VRR2(bt) homeomorphic to VRR2(a1), …, VRR2(as) behave when t approaches zero? They are moving to the coordinate axes and infinity. The closer t to zero, the more place is occupied by the domain, where VRR2(bt) is organized as VRR2(a1) and is approximated by it (cf. Section 6.7).

It is curious that the family bt can be changed by a simple geometric transformation in such a way that the role of a1 passes to any one of a2,,as or even to akΓ, where Γ is a side of Δ(ak), k=1,,s. Indeed, let λ:R2R be a linear function, λ(x,y)=αx+βy+γ. Let ν=νλ. Denote by bt the result of patchworking a1,,as by ν. Denote by qh(a,b),t the linear transformation RR2RR2:(x,y)(xta,ytb). Then

VRR2(bt)=VRR2(btqh(α,β),t)=qh(α,β),tVRR2(bt).

Indeed,

bt(x,y)= aωxω1yω2tν(ω1,ω2)αω1βω2γ
=tγaω(xtα)ω1(ytβ)ω2tν(ω1,ω2)
=tγbt(xtα,ytβ)
=tγbtqh(α,β),t(x,y).

Thus the curves VRR2(bt) and VRR2(bt) are transformed to each other by a linear transformation. However the polynomial bt does not tend to a1 as t0. For example, if λ|Δ(ak)=ν|Δ(ak) then ν|Δ(ak)=0 and btak. In this case as t0, the domains containing parts of VRR2(bt), which are homeomorphic to VRR2(ai), with ik, run away and the domain in which VRR2(bt) looks like VRR2(ak) occupies more and more place. If the set, where ν coincides with λ (or differs from λ by a constant), is a side Γ of Δ(ak), then the curve VRR2(bt) turns to VRR2(akΓ) (i.e. collection of quasilines) as t0 similarly.

The whole picture of evolution of VRR2(bt) when t0 is the following. The fragments which look as VRR2(ai) with i=1,,s become more and more explicit, but these fragments are not staying. Each of them is moving away from the others. The only fragment that is growing without moving corresponds to the set where ν is constant. The other fragments are moving away from it. From the metric viewpoint some of them (namely, ones going to the origin and axes) are contracting, while the others are growing. But in the logarithmic coordinates, i.e. being transformed by l:(x,y)(ln|x|,ln|y|), all the fragments are growing (see Section 6.7). Changing ν we are applying linear transformation, which distinguishes one fragment and casts away the others. The transformation turns our attention to a new piece of the curve. It is as if we would transfer a magnifying lens from one fragment of the curve to another. Naturally, under such a magnification the other fragments disappear at the moment t=0.

1.10. Patchworking as smoothing of singularities

In the projective plane the passage from curves defined by bt with t>0 to the curve defined by b0 looks quite differently. Here, the domains, in which the curve defined by bt looks like curves defined by a1,,as are not running away, but pressing more closely to the points (1:0:0), (0:1:0), (0:0:1) and to the axes joining them. At t=0, they are pressed into the points and axes. It means that under the inverse passage (from t=0 to t>0) the full or partial smoothing of singularities concentrated at the points (1:0:0), (0:1:0), (0:0:1) and along coordinate axes happens.

Figure 21.

Figure 22.
 Example 1.10.A  .

Let a1, a2 be polynomials of degree 6 with a1Δ(a1)Δ(a2)=a2Δ(a1)Δ(a2) and charts shown in Figure 1.8 (a) and 1.8 (b). Let ν1, ν2 and ν3 be defined by the following formulas:

ν1(ω1,ω2) ={0,if ω1+2ω262(ω1+2ω26),if ω1+2ω26
ν2(ω1,ω2) ={6ω12ω2,if ω1+2ω26ω1+2ω26,if ω1+2ω26
ν3(ω1,ω2) ={2(6ω12ω2),if ω1+2ω260,if ω1+2ω26

(note, that ν1, ν2 and ν3 differ from each other by a linear function). Let bt1, bt2 and bt3 be the results of patchworking a1, a2 by ν1, ν2 and ν3. By Theorem 1.7.A  for sufficiently small t>0 the polynomials bt1, bt2 and bt3 have the same chart shown in Figure 1.8 (ab), but as t0 they go to different polynomials, namely, a1, a1Δ(a1)Δ(a2) and a2.The closure of VRR2(bti) with i=1, 2, 3 in the projective plane (they are transformed to one another by projective transformations) are shown in Figure Figure 21. The limiting projective curves, i.e. the projective closures of VRR2(a1), VRR2(a1Δ(a1)Δ(a2)), VRR2(a2) are shown in Figure 22. The curve shown in Figure 22 (b) is the union of three nonsingular conics which are tangent to each other in two points.

Curves of degree 6 with eleven components of all three isotopy types can be obtained from this curve by small perturbations of the type under consideration (cf. Section 1.8). Moreover, as it is proven in [Vir89], Section 5.1, nonsingular curves of degree 6 of almost all isotopy types can be obtained.

1.11. Evolvings of singularities

Let f be a real polynomial in two variables. (See Section 5, where more general situation with an analytic function playing the role of f is considered.) Suppose its Newton polygon Δ(f) intersects both coordinate axes (this assumption is equivalent to the assumption that VR2(f) is the closure of VRR2(f)). Let the distance from the origin to Δ(f) be more than 1 or, equivalently, the curve VR2(f) has a singularity at the origin. Let this singularity be isolated. Denote by B a disk with the center at the origin having sufficiently small radius such that the pair (B,VB(f)) is homeomorphic to the cone over its boundary (B,VB(f)) and the curve VR2(f) is transversal to B (see [Mil68], Theorem 2.10).

Let f be included into a continuous family ft of polynomials in two variables: f=f0. Such a family is called a perturbation of f. We shall be interested mainly in perturbations for which curves VR2(ft) have no singular points in B when t is in some segment of type (0,ε]. One says about such a perturbation that it evolves the singularity of VR2(ft) at zero. If perturbation ft evolves the singularity of VR2(f) at zero then one can find t0>0 such that for t(0,t0] the curve VR2(ft) has no singularities in B and, moreover, is transversal to B. Obviously, there exists an isotopy ht:BB with t0(0,t0] such that ht0=id and ht(VB(f0))=VB(ft), so all pairs (B,VB(ft)) with t(0,t0] are homeomorphic to each other. A family (B,VR2(ft)) of pairs with t(0,t0] is called an evolving of singularity of VR2(f) at zero, or an evolving of germ of VR2(f).

Denote by Γ1,,Γn the sides of Newton polygon Δ(f) of the polynomial f, faced to the origin. Their union Γ(f)=i=1nΓi is called the Newton diagram of f.

Suppose the curves VRR2(fΓi) with i=1,,n are nonsingular. Then, according to Newton [New67], the curve VR2(f) is approximated by the union of ClVRR2(fΓi) with i=1,,n in a sufficiently small neighborhood of the origin. (This is a local version of Theorem 1.2.A  ; it is, as well as 1.2.A  , a corollary of Theorem 6.3.A  .) Disk B can be taken so small that VB(f) is close to BVRR2(fΓi), so the number and disposition of these points are defined by charts (Γi,υi) of fiΓ. The union (Γ(f),υ)=(i=1nΓi,i=1nυi) of these charts is called a chart of germ of VR2(f) at zero. It is a pair consisting of a simple closed polygon Γ(f), which is symmetric with respect to the axes and encloses the origin, and finite set υ lying on it. There is a natural bijection of this set to VB(f), which is extendable to a homeomorphism (Γ(f), υ)(B,VB(f)). Denote this homeomorphism by g.

Let ft be a perturbation of f, which evolves the singularity at the origin. Let B, t0 and ht be as above. It is not difficult to choose an isotopy ht:BB, t(0,t0] such that its restriction to B can be extended to an isotopy ht:BB with t[0,t0] and h0(VB(ft0))=VB(f). A pair (Π,τ) consisting of the polygon Π bounded by Γ(f) and an 1-dimensional subvariety τ of Π is called a chart of evolving (B,VB(ft)), t[0,t0] if there exists a homeomorphism ΠB, mapping τ to VB(ft0), whose restriction ΠΠ is the composition Γ(f)@>g>>B@>h0>>B. It is clear that the boundary (Π,τ) of a chart of germ’s evolving is a chart of the germ. Also it is clear that if polynomial f is completely nondegenerate and polygons Δ(ft) are obtained from Δ(f) by adjoining the region restricted by the axes and Π(f), then charts of ft with t(0,t0] can be obtained by patchworking a chart of f and chart of evolving (B,VB(ft)), t[0,t0].

The patchworking construction for polynomials gives a wide class of evolvings whose charts can be created by the modification of Theorem 1.7.A  formulated below.

Let a1,,as be completely nondegenerate polynomials in two variables with IntΔ(ai)IntΔ(aj)= and aiΔ(ai)Δ(aj)=ajΔ(ai)Δ(aj) for ij. Let i=1sΔ(ai) be a polygon bounded by the axes and Γ(f). Let aiΔ(ai)Δ(f)=fΔ(ai)Δ(f) for i=a,,s. Let ν:R2R be a nonnegative convex function which is equal to zero on Δ(f) and whose restriction on i=1sΔ(ai) satisfies the conditions 1, 2 and 3 of Section 1.6 with respect to a1,,as. Then a result ft of patchworking f, a1,,as by ν is a perturbation of f.

Theorem 1.7.A  cannot be applied in this situation because the polynomial f is not supposed to be completely nondegenerate. This weakening of assumption implies a weakening of conclusion.

1.11.A  Local version of Theorem 1.7.A  .

Under the conditions above perturbation ft of f evolves a singularity of VR2(f) at the origin. A chart of the evolving can be obtained by patchworking charts of a1,,as.

An evolving of a germ, constructed along the scheme above, is called a patchwork evolving.

If Γ(f) consists of one segment and the curve VRR2(fΓ(f)) is nonsingular then the germ of VR2(f) at zero is said to be semi-quasi-homogeneous. In this case for construction of evolving of the germ of VR2(f) according the scheme above we need only one polynomial; by 1.11.A  , its chart is a chart of evolving. In this case geometric structure of VB(ft) is especially simple, too: the curve VB(ft) is approximated by qhw,t(VR2(a1)), where w is a vector orthogonal to Γ(f), that is by the curve VR2(a1) contracted by the quasihomothety qhw,t. Such evolvings were described in my paper [Vir80]. It is clear that any patchwork evolving of semi-quasi-homogeneous germ can be replaced, without changing its topological models, by a patchwork evolving, in which only one polynomial is involved (i.e. s=1).

2. Toric varieties and their hypersurfaces

2.1. Algebraic tori KRn

In the rest of this chapter K denotes the main field, which is either the real number field R, or the complex number field C.

For ω=(ω1,,ωn)Zn and ordered collection x of variables x1,,xn the product x1ω1xnωn is denoted by xω. A linear combination of products of this sort with coefficients from K is called a Laurent polynomial or, briefly, L-polynomial over K. Laurent polynomials over K in n variables form a ring K[x1,x11,,xn,xn1] naturally isomorphic to the ring of regular functions of the variety (K0)n.

Below this variety, side by side with the affine space Kn and the projective space KPn, is one of the main places of action. It is an algebraic torus over K. Denote it by KRn.

Denote by l the map KRnRn defined by formula l(x1,,xn)= (ln|x1|, , ln|xn|).

Put UK={xK||x|=1}, so UR=S0 and UC=S1. Denote by ar the map KRnUKn (=UK××UK) defined by ar(x1,,xn)=(x1|x1|,,xn|xn|).

Denote by la the map

x(l(x),ar(x)):KRnRn×UKn.

It is clear that this is a diffeomorphism.

KRn is a group with respect to the coordinate-wise multiplication, and l, ar, la are group homomorphisms; la is an isomorphism of KRn to the direct product of (additive) group Rn and (multiplicative) group UKn.

Being Abelian group, KRn acts on itself by translations. Let us fix notations for some of the translations involved into this action.

For wRn and t>0 denote by qhw,t and call a quasi-homothety with weights w=(w1,,wn) and coefficient t the transformation KRnKRn defined by formula qhw,t(x1,,xn)=(tw1x1,,twnxn), i.e. the translation by (tw1,,twn). If w=(1,,1) then it is the usual homothety with coefficient t. It is clear that qhw,t=qhλ1w,t for λ>0. Denote by qhw a quasi-homothety qhw,e, where e is the base of natural logarithms. It is clear, qhw,t=qh(lnt)w.

For w=(w1,,wn)UKn denote by Sw the translation KRnKRn defined by formula

Sw(x1,,xn)=(w1x1,,wnxn),

i. e. the translation by w.

For wRn denote by Tw the translation xx+w:RnRn by the vector w.

2.1.A  .

Diffeomorphism la:KRnRn×UKn transforms qhw,t to T(lnt)w×idUKn, and Sw to idRn×(Sw|UKn), i.e.

laqhw,tla1=T(lnt)w×idUKnand
laSwla1=idRn×(Sw|UKn).

In particular, laqhwla1=Tw×id.

A hypersurface of KRn defined by a(x)=0, where a is a Laurent polynomial over K in n variables is denoted by VKRn(a).

If a(x)=ωZnaωxω is a Laurent polynomial, then by its Newton polyhedron Δ(a) is the convex hull of {ωRn|aω0}.

2.1.B  .

Let a be a Laurent polynomial over K. If Δ(a) lies in an affine subspace Γ of Rn then for any vector wRn orthogonal to Γ, a hypersurface VKRn(a) is invariant under qhw,t.

Proof.

Since Δ(a)Γ and Γw, then for ωΔ(a) the scalar product wω does not depend on ω. Hence

a(qhw,t1(x))=ωΔ(a)aω(twx)ω=twωωΔ(a)aωxω=twωa(x),

and therefore

qhw,t(VKRn(a))=VKRn(aqhw,t1)=VKRn(twωa)=VKRn(a).

Proposition 2.1.B  is equivalent, as it follows from 2.1.A  , to the assertion that under hypothesis of 2.1.B  the set la(VKRn(a)) contains together with each point (x,y)Rn×UKn all points (x,y)Rn×UKn with xxΓ. In other words, in the case Δ(a)Γ the intersections of la(VKRn(a)) with fibers Rn×y are cylinders, whose generators are affine spaces of dimension ndimΓ orthogonal to Γ.

The following proposition can be proven similarly to 2.1.B  .

2.1.C  .

Under the hypothesis of 2.1.B  a hypersurface VKRn(a) is invariant under transformations S(eπiw1,,eπiwn), where wΓ,

w{Zn,\, if K=RRn,\, if K=C.

In other words, under the hypothesis of 2.1.B  the hypersurface VKRn(a) contains together with each its point (x1,,xn):

  1. (1)

    points ((1)w1x1,,(1)wnxn) with wZn, wΓ, if K=R,

  2. (2)

    points (eiw1x1,,eiwnxn) with wRn, wΓ, if K=C.

2.2. Polyhedra and cones

Below by a polyhedron we mean closed convex polyhedron lying in Rn, which are not necessarily bounded, but have a finite number of faces. A polyhedron is said to be integer if on each of its faces there are enough points with integer coordinates to define the minimal affine space containing this face. All polyhedra considered below are assumed to be integer, unless the contrary is stated.

The set of faces of a polyhedron Δ is denoted by G(Δ), the set of its k-dimensional faces by Gk(Δ), the set of all its proper faces by G(Δ).

By a halfspace of vector space V we will mean the preimage of the closed halfline R+(={xR:x0}) under a non-zero linear functional VR (so the boundary hyperplane of a halfspace passes necessarily through the origin). By a cone it is called an intersection of a finite collection of halfspaces of Rn. A cone is a polyhedron (not necessarily integer), hence all notions and notations concerning polyhedra are applicable to cones.

The minimal face of a cone is the maximal vector subspace contained in the cone. It is called a ridge of the cone.

For v1,,vkRn denote by v1,,vk the minimal cone containing v1, …, vk; it is called the cone generated by v1,,vk. A cone is said to be simplicial if it is generated by a collection of linear independent vectors, and simple if it is generated by a collection of integer vectors, which is a basis of the free Abelian group of integer vectors lying in the minimal vector space which contains the cone.

Let ΔRn be a polyhedron and Γ its face. Denote by CΔ(Γ) the cone rR+r(Δy), where y is a point of ΓΓ. The cone CΔ(Δ) is clearly the vector subspace of Rn which corresponds to the minimal affine subspace containing Δ. The cone CΓ(Γ) is the ridge of CΔ(Γ). If Γ is a face of Δ with dimΓ=dimΔ1, then CΔ(Γ) is a halfspace of CΔ(Δ) with boundary parallel to Γ.

For cone CRn we put

D+C={xRn|aCax0},
DC={xRn|aCax0}.

These are cones, which are said to be dual to C. The cones D+C and DC are symmetric to each other with respect to 0. The cone DC permits also the following more geometric description. Each hyperplane of support of C defines a ray consisting of vectors orthogonal to this hyperplane and directed to that of two open halfspaces bounded by it, which does not intersect C. The union of all such rays is DC.

It is clear that D+D+C=C=DDC. If v1,,vn is a basis of Rn, then the cone D+v1,,vn is generated by dual basis v1,,vn (which is defined by conditions vivj=Δij).

2.3. Affine toric variety

Let ΔRn be an (integer) cone. Consider the semigroup K-algebra K[ΔZn] of the semigroup ΔZn. It consists of Laurent polynomials of the form ωΔZnaωxω. According to the well known Gordan Lemma (see, for example, [Dan78], 1.3), the semigroup ΔZn is generated by a finite number of elements and therefore the algebra K[ΔZn] is generated by a finite number of monomials. If this number is greater than the dimension of Δ, then there are nontrivial relations among the generators; the number of relations of minimal generated collection is equal to the difference between the number of generators and the dimension of Δ.

An affine toric variety KΔ is the affine scheme SpecK[ΔZn]. Its less invariant, but more elementary definition looks as follows. Let

{α1,,αp|i=1pu1,iαi=i=1pvi,1αi,,i=1pupn,iαi=i=1pvpn,iαi}

be a presentation of ΔZn by generators and relations (here uij and vij are nonnegative); then the variety KΔ is isomorphic to the affine subvariety of Kp defined by the system

{y1u11ypu1p=y1v11ypv1py1upn,1ypupn,p=y1vpn,1ypvpn,p.

For example, if Δ=Rn, then KΔ=SpecK[x1,x11,,xn,xn1] can be presented as the subvariety of K2n defined by the system

{y1yn+1=1yny2n=1

Projection K2nKn induces an isomorphism of this subvariety to (K0)n=KRn. This explains the notation KRn introduced above.

If Δ is the positive orthant An={xRn|x10,,xn0}, then KΔ is isomorphic to the affine space Kn. The same takes place for any simple cone. If cone is not simple, then corresponding toric variety is necessarily singular. For example, the angle shown in Figure 23 corresponds to the cone defined in K3 by xy=z2.

Figure 23.

Let a cone Δ1 lie in a cone Δ2. Then the inclusion in:Δ1Δ2 defines an inclusion K[Δ1Zn]K[Δ2Zn] which, in turn, defines a regular map

in:SpecK[Δ2Zn]SpecK[Δ1Zn],

i.e. a regular map in:KΔ2KΔ1. The latter can be described in terms of subvarieties of affine spaces in the following way. The formulas, defining coordinates of point in(y) as functions of coordinates of y, are the multiplicative versions of formulas, defining generators of semigroup Δ1Zn as linear combinations of generators of the ambient semigroup Δ2Zn.

In particular, for any Δ there is a regular map of KCΔ(Δ)KRdimΔ to KΔ. It is not difficult to prove that it is an open embedding with dense image, thus KΔ can be considered as a completion of KRdimΔ.

An action of algebraic torus KCΔ(Δ) in itself by translations is extended to its action in KΔ. This extension can be obtained, for example, in the following way. Note first, that for defining an action in KΔ it is sufficient to define an action in the ring K[ΔZn]. Define an action of KRn on monomials xωK[δZn] by formula (α1,,αn)xω=α1ω1αnωn and extend it to the whole ring K[ΔZn] by linearity. Further, note that if VRn is a vector space, then the map in:KRnKV is a group homomorphism. Elements of kernel of in:KRnKCΔ(Δ) act identically in K[ΔZn]. It allows to extract from the action of KRn in KΔ an action of KCΔ(Δ) in KΔ, which extends the action of KCΔ(Δ) in itself by translations.

With each face Γ of a cone Δ one associates (as with a smaller cone) a variety KΓ and a map in:KΔKΓ. On the other hand there exists a map in:KΓKΔ for which inin is the identity map KΓKΓ. Therefore, in is an embedding whose image is a retract of KΔ. From the viewpoint of schemes the map in should be defined by the homomorphism K[ΔZn]K[ΓZn] which maps a Laurent polynomial ωΔZnaωxω to its Γ-truncation ωΓZnaωxω. In terms of subvarieties of affine space, KΓ is the intersection of KΔ with the subspace yi1=yi2==yis=0, where yi1,,yis are the coordinates corresponding to generators of semigroup ΔZn which do not lie in Γ.

Varieties in(KΓ) with ΓGdimΔ1(Δ) cover KΔin(KCΔ(Δ)). Images of algebraic tori KCΓ(Γ) with ΓG(Δ) under the composition

\begin{CD}KC_{\Gamma}(\Gamma)@>{\operatorname{in}^{*}}>{}>K\Gamma @>{%\operatorname{in}^{*}}>{}>K\Delta\end{CD}

of embeddings form a partition of KΔ, which is a smooth stratification of KΔ. Closure of the stratum inin(KCΓ(Γ)) in KΔ is in(KΓ). Below in the cases when it does not lead to confusion we shall identify KΓ with inKΓ and KCΓ(Γ) with ininKCΓ(Γ) (i.e. we shall consider KΓ and KCΓ(Γ) as lying in KΔ).

2.4. Quasi-projective toric variety

Let ΔRn be a polyhedron. If Γ is its face and Σ is a face of Γ, then CΓ(Σ) is a face of CΔ(Γ) parallel to Γ, and CCΔ(Σ)(CΓ(Σ))=CΔ(Γ), see Figure 24.

Figure 24.

In particular, CΔ(Σ)CΔ(Γ) and, hence, the map in:KCΔ(Γ)KCΔ(Σ) is defined. It is easy to see that this is an open embedding. Let us glue all KCΔ(Γ) with ΓG(Δ) together by these embeddings. The result is denoted by KΔ and called the toric variety associated with Δ. This definition agrees with the corresponding definition from the previous Section: if Δ is a cone and Σ is its ridge then CΔ(Σ)=Δ and, since the ridge is the minimal face, all KCΔ(Γ) with ΓG(Δ) are embedded in KCΔ(Σ) and the gluing gives KCΔ(Σ)=KΔ.

For any polyhedron Δ the toric variety KΔ is quasi-projective. If Δ is bounded, it is projective (see [GK73] and [Dan78]).

A polyhedron ΔRn is said to be permissible if dimΔ=n, each face of Δ has a vertex and for any vertex ΓG0(Δ) the cone CΔ(Γ) is simple. If polyhedron Δ is permissible then variety KΔ is nonsingular and it can be obtained by gluing affine spaces KCΔ(Γ) with ΓG0(Δ). The gluing allows the following description. Let us associate with each cone CΔ(Γ) where ΓG0(Δ) an automorphism fΓ:KRnKRn: if CΔ(Γ)=v1,,vn and vi=(vi1,,vin) for i=1,,n, then we put fΓ(x1,,xn)=(x1v11xnv1n,,x1vn1xnvnn). The variety KΔ is obtained by gluing to KRn copies of Kn by maps \begin{CD}K{\mathbb{R}}^{n}@>{f_{\Gamma}}>{}>K{\mathbb{R}}^{n}\hookrightarrow K%^{n}\end{CD} for all vertices Γ of Δ. (Cf. Khovansky [Kho77].)

The variety KΔ is defined by Δ, but does not define it. Indeed, if Δ1 and Δ2 are polyhedra such that there exists a bijection G(Δ1)G(Δ2), preserving dimensions and inclusions and assigning to each face of Δ1 a parallel face of Δ2, then KΔ1=KΔ2.

Denote by Pn the simplex of dimension n with vertices

(0,0,,0),(1,0,,0),(0,1,0,,0),,(0,0,,1).

It is permissible polyhedron. KPn is the n-dimensional projective space (this agrees with its usual notation).

Evidently, K(Δ1×Δ2)=KΔ1×KΔ2. In particular, if ΔR2 is a square with vertices (0,0), (1,0), (0,1) and (1,1), i.e. if Δ=P1×P1, then KΔ is a surface isomorphic to nonsingular projective surface of degree 2 (to hyperboloid in the case of K=R2).

Polyhedra shown in Figure 25 define the following surfaces: KΔ1 is the affine plane with a point blown up; KΔ2 is projective plane with a point blown up (RΔ2 is the Klein bottle); KΔ3 is the linear surface over KP1, defined by sheaf O+O(2) (RΔ3 is homeomorphic to torus).

Figure 25.

The variety KCΔ(Δ) is isomorphic to KRdimΔ, open and dense in KΔ, so KΔ can be considered as a completion of KRdimΔ. Actions of KCΔ(Δ) in affine parts KCΔ(Γ) of KΔ correspond to each other and define an action in KΔ which is an extension of the action of KCΔ(Δ) in itself by translations. Transformations of KΔ extending qhw,t and Sw are denoted by the same symbols qhw,t and Sw.

The complement KΔKCΔ(Δ) is covered by KΣ with ΣG(CΔ(Γ)), ΓG(Δ) or, equivalently, by varieties KCΓ(Σ) with ΣG(Γ), ΓG(Δ). They comprise varieties KΓ with ΓG(Δ), which also cover KΔKCΔ(Δ). The varieties KΓ are situated with respect to each other in the same manner as the corresponding faces in the polyhedron: K(Γ1Γ2)=KΓ1KΓ2. Algebraic tori KCΓ(Γ)=KΓΣG(Γ)KΣ form partition of KΔ, which is a smooth stratification; they are orbits of the action of KCΔ(Δ) in KΔ.

We shall say that a polyhedron Δ2 is richer than a polyhedron Δ1 if for any face Γ2G(Δ2) there exists a face Γ1G(Δ1) such that CΔ2(Γ2)CΔ1(Γ1) (such a face Γ1 is automatically unique), and for each face Γ1G(Δ1) the cone CΔ1(Γ1) can be presented as the intersection of several cones CΔ2(Γ2) with Γ1G(Δ2). This definition allows a convenient reformulation in terms of dual cones: a polyhedron Δ2 is richer than polyhedron Δ1 iff the cones D+CΔ2(Γ2) with Γ2G(Δ2) cover the set, which is covered by D+CΔ1(Γ1) with Γ1G(Δ1), and the first covering is a refinement of the second.

Let a polyhedron Δ2 be richer than Δ1. Then the inclusions CΔ1(Γ1)CΔ2(Γ2) define for any Γ2G(Δ2) a regular map \begin{CD}KC_{\Delta_{2}}(\Gamma_{2})@>{\operatorname{in}^{*}}>{}>KC_{\Delta_{%1}}(\Gamma_{1})\hookrightarrow K\Delta_{1}\end{CD}. Obviously, these maps commute with the embeddings, by which KΔ2 and KΔ1 are glued from affine pieces, thus a regular map KΔ2KΔ1 appears.

One can show (see, for example, [GK73]) that for any polyhedron Δ1 there exists a richer polyhedron Δ2, defining a nonsingular toric variety KΔ2. Such a polyhedron is called a resolution of Δ1 (because it gives a resolution of singularities of KΔ1). If dimΔ=n (= the dimension of the ambient space Rn), then a resolution of Δ can be found among permissible polyhedra.

2.5. Hypersurfaces of toric varieties

Let ΔRn be a polyhedron and a be a Laurent polynomial over K in n variables. Let CΔ(a)(Δ(a))CΔ(Δ). Then there exists a monomial xω such that Δ(xωa)CΔ(Δ). The hypersurface VKCΔ(Δ) does not depend on the choice of xω and is denoted simply by VKCΔ(Δ)(a). Its closure in KΔ is denoted by VKΔ(a). 44Here it is meant the closure of KΔ in the Zarisky topology; in the case of K=C the classic topology gives the same result, but in the case of K=R the usual closure may be a nonalgebraic set. Thus, to any Laurent polynomial a over K with CΔ(a)(Δ(a))CΔ(Δ), a hypersurface VKΔ(a) of KΔ is related. For Laurent polynomial a(x)=ωZnaωxω and a set ΓRn a Laurent polynomial a(x)=ωΓZnaωxω is denoted by aΓ and called the Γ-truncation of a.

2.5.A  .

Let ΔRn be a polyhedron and a be a Laurent polynomial over K with CΔ(a)(Δ(a))CΔ(Δ). If Γ1G(Δ(a)), Γ2G(Δ) and CΔ(a)(Γ1)CΔ(Γ2) then KΓ2VKΔ(a)=VKΓ2(aΓ1).

Proof.

Consider KCΔ(Γ2). It is a dense subset of KΓ2. Since CΔ(a)(Γ1)CΔ(Γ2), there exists a monomial xω such that Δ(xωa) lies in CΔ(Γ2) and intersects its ridge exactly in the face obtained from Γ1. Since on KΓ2KCΔ(Γ2) all monomials, whose exponents do not lie on ridge CΓ2(Γ2) of CΔ(Γ2), equal zero, it follows that the intersection {xKCΔ(Γ2)|xωa(x)=0}KΓ2 coincides with {xKCΔ(Γ2)|[xωa]CΓ2(Γ2)(x)=0}KΓ2. Note finally, that the latter coincides with VKΓ2(a1).

2.5.B  .

Let Δ and a be as in 2.5.A  and Γ2 be a proper face of the polyhedron Δ. If there is no face Γ1G(Δ(a)) with CΔ(a)(Γ1)CΔ(Γ2) then KΓ2VKΔ(a).

The proof is analogous to the proof of the previous statement.

Denote by SVKRn(a) the set of singular points of VKRn(a), i.e. a set VKRn(a)i=1nVKRn(axi).

A Laurent polynomial a is said to be completely nondegenerate (over K) if, for any face Γ of its Newton polyhedron, SVKRn(aΓ) is empty and, hence, VKRn(aΓ) is a nonsingular hypersurface. A Laurent polynomial a is said to be peripherally nondegenerate if for any proper face Γ of its Newton polyhedron SVKRn(aΓ)=.

It is not difficult to prove that completely nondegenerate L-polynomials form Zarisky open subset of the space of L-polynomials over K with a given Newton polyhedron, and the same holds true also for peripherally nondegenerate L-polynomials.

2.5.C  .

If a Laurent polynomial a over K is completely nondegenerate and ΔRn is a resolution of its Newton polyhedron Δ(a) then the variety VKΔ(a) is nonsingular and transversal to all KΓ with ΓG(Δ). See, for example, [Kho77].

Theorem 2.5.C  allows various generalizations related with possibilities to consider singular KΔ or only some faces of Δ(a) (instead of all of them). For example, one can show that if under the hypothesis of 2.5.A  a truncation aΓ of a is completely nondegenerate then under an appropriate understanding of transversality (in the sense of stratified space theory) VKΔ(a) is transversal to KΓ2. Without going into discussion of transversality in this situation, I formulate a special case of this proposition, generalizing Theorem 2.5.C  .

2.5.D  .

Let Γ be a face of a polyhedron ΔRn with nonempty G0(Γ) and with simple cones CΔ(Σ) for all ΣG0(Γ). Let a be a Laurent polynomial over K in n variables and Γ1 be a face of Δ(a) with CΔ(a)(Γ1)CΔ(Γ). If aΓ is completely nondegenerate, then the set of singular points of VKΔ(a) does not intersect KΓ and VKΔ(a) is transversal to KΓ.

The proof of this proposition is a fragment of the proof of Theorem 2.5.C  .

2.5.E  (Corollary of 2.1.B  and 2.1.C  ).

Let Δ and a be as in 2.5.A  . Then for any vector wCΔ(Δ) orthogonal to CΔ(a)(Δ(a)), a hypersurface VKΔ(a) is invariant under transformations qhw,t:KΔKΔ and S(eπiw1,,eπiwn):KΔKΔ (the latter in the case of K=R is defined only if wZn).

3. Charts

3.1. Space R+Δ

The aim of this Subsection is to distinguish in KΔ an important subspace which looks like Δ. More precisely, it is defined a stratified real semialgebraic variety R+Δ, which is embedded in KΔ and homeomorphic, as a stratified space, to the polyhedron Δ stratified by its faces. Briefly R+Δ can be described as the set of points with nonnegative real coordinates.

If Δ is a cone then R+Δ is defined as a subset of KΔ consisting of the points in which values of all monomials xω with ωΔZn are real and nonnegative. It is clear that for ΓG(Δ) the set R+Γ coincides with R+ΔKΓ and for cones Δ1Δ2 a preimage of R+Δ1 under in:KΔ2KΔ1 (see Section 2.3) is R+Δ2.

Now let Δ be an arbitrary polyhedron. Embeddings, by which KΔ is glued form KCΔ(Γ) with ΓG(Δ), embed the sets R+CΔ(Γ) in one another; a space obtained by gluing from R+CΔ(Γ) with ΓG(Δ) is R+Δ. It is clear that if ΓG then R+Γ=R+ΔKΓ.

R+Rn is the open positive orthant {xRRn|x1>0,,xn>0}. It can be identified with the subgroup of quasi-homotheties of KRn: one assigns qhl(x) to a point xR+Rn.

If An={xRn|x10,,xn0} then KAn=Kn (cf. Section 2.3) and R+An=An.

If Pn is the n-simplex with vertexes (0,0,,0), (1,0,,0), (0,1,0,,0), …, (0,0,,1), then KPn is the n-simplex consisting of points of projective space with nonnegative real homogeneous coordinates.

The set R+Δ is invariant under quasi-homotheties. Orbits of action in R+Δ of the group of quasi-homotheties of R+Rn are sets R+CΓ(Γ) with ΓG(Δ). Orbit R+CΓ(Γ) is homeomorphic to RdimΓ or, equivalently, to the interior of Γ. Closures R+Γ of R+CΓ(Γ) intersect one another in the same manner as the corresponding faces: R+Γ1R+Γ2=R+(Γ1Γ2). From this and from the fact that R+Γ is locally conic (see [Loj64]) it follows that R+Δ is homeomorphic, as a stratified space, to Δ. However, there is an explicitly constructed homeomorphism. It is provided by the Atiyah moment map [Ati81] and in the case of bounded Δ can be described in the following way.

Choose a collection of points ω1,,ωk with integer coordinates, whose convex hull is Δ. Then for ΓG(Δ) and ω0ΓΓ cone CΔ(Γ) is ω1ω0,,ωkω0. For yKCΔ(Γ) denote by yω a value of monomial xω where ωCΔ(Γ)Zn at this point. Put

M(y)=i=1k|yωiω0|ωii=1k|yωiω0|Rn.

Obviously M(y) lies in Δ, does non depend on the choice of ω0 and for yKCΔ(Γ1)KCΔ(Γ2) does not depend on what face, Γ1 or Γ2, is used for the definition of M(y). Thus a map M:KΔΔ is well defined. It is not difficult to show that M|R+Δ:R+ΔΔ is a stratified homeomorphism.

3.2. Charts of KΔ

The space KRn can be presented as R+Rn×UKn. In this Section an analogous representation of KΔ is described.

R+Δ is a fundamental domain for the natural action of UKn in KΔ, i.e. its intersection with each orbit of the action consists of one point.

For a point xR+CΓΓ where ΓG(Δ), the stationary subgroup of action of UKn consists of transformations S(eπiw1,,eπiwn), where vector (w1,,wn) is orthogonal to CΓ(Γ). In particular, if dimΓ=n then the stationary subgroup is trivial. If dimΓ=nr then it is isomorphic to UKr. Denote by UΓ a subgroup of UKn consisting of elements (eπiw1,,eπiwn) with (w1,,wn)CΓ(Γ).

Define a map ρ:R+Δ×UKnKΔ by formula (x,y)Sy(x). It is surjection and we know the partition of R+Δ×UKn into preimages of points. Since ρ is proper and KΔ is locally compact and Hausdorff, it follows that KΔ is homeomorphic to the quotientspace of R+Δ×UKn with respect to the partition into sets x×yUΓ with xR+CΓ(Γ), yUKn.

Consider as an example the case of K=R and n=2. Let a polyhedron Δ lies in the open positive quadrant. We place Δ×UR2 in R2 identifying (x,y)Δ×UR2 with Sy(x)R2. R+Δ×UR2 is homeomorphic Δ×UR2, so the surface RΔ can be obtained by an appropriate gluing (namely, by transformations taken from UΓ) sides of four polygons consisting Δ×UR2. Figure 26 shows what gluings ought to be done in three special cases.

Figure 26.

3.3. Charts of L-polynomials

Let a be a Laurent polynomial over K in n variables and Δ be its Newton polyhedron. Let h be a homeomorphism ΔR+Δ, mapping each face to the corresponding subspace, and such that for any ΓG(Δ), xΓ, yUKn, zUΓ

h(x,y,z)=(prR+Γh(x,y),zprUKnh(x,y)).

For h one can take, for example, (M|R+Δ).

A pair consisting of Δ×UKn and its subset υ which is the preimage of VKΔ(a) under

\begin{CD}\Delta\times U_{K}^{n}@>{h\times\operatorname{id}}>{}>{\mathbb{R}}_{%+}\Delta\times U_{K}^{n}@>{\rho}>{}>K\Delta\end{CD}

is called a (nonreduced) K-chart of L-polynomial a.

It is clear that the set υ is invariant under transformations id×S with SUΔ and its intersection with Γ×UKn, where ΓG(Δ) is invariant under transformations id×S with SUΓ.

As it follows from 2.5.A  , if Γ is a face of Δ, and (Δ×UKn,υ) is a nonreduced K-chart of L-polynomial a, then (Γ×UKn,υ(Γ×UKn)) is a nonreduced K-chart of L-polynomial aΓ.

A nonreduced K-chart of Laurent polynomial a is unique up to homeomorphism Δ×UKnΔ×UKn, satisfying the following two conditions:

  1. (1)

    it map Γ×y with ΓG(Δ) and yUKn to itself and

  2. (2)

    its restriction to Γ×UKn with gG(Δ) commutes with transformations id×S:Γ×UKnΓ×UKn where SUΓ.

In the case when a is a usual polynomial, it is convenient to place its K-chart into Kn. For this, consider a map An×UKnKn:(x,y)Sy(x). Denote by ΔK(a) the image of Δ(a)×UKn under this map. Call by a (reduced) K-chart of a the image of a nonreduced K-chart of a under this map. The charts of peripherally nondegenerate real polynomial in two variables introduced in Section 1.3 are R-charts in the sense of this definition.

3.3.A  .

Let a be a Laurent polynomial over K in n variables, Γ a face of its Newton polyhedron, ρ:R+Δ(a)×UKnKΔ(a) a natural projection. If the truncation aΓ is completely nondegenerate then the set of singular points of hypersurface ρ1VKΔ(a)(a) of  R+Δ(a)×UKn does not intersect R+Γ×UKn, and ρ1VKΔ(a)(a) is transversal to R+Γ×UKn.

Proof.

Let Δ be a resolution of polyhedron Δ(a). Then a commutative diagram

\begin{CD}({\mathbb{R}}_{+}\Delta\times U_{K}^{n},\,\rho^{\prime-1}(V_{K\Delta%}(a)))@>{\rho^{\prime}}>{}>(K\Delta,V_{K\Delta}(a))\\@V{({\mathbb{R}}_{+}s\times\operatorname{id})}V{}V@V{s}V{}V\\({\mathbb{R}}_{+}\Delta(a)\times U_{K}^{n},\,\rho^{-1}(V_{K\Delta(a)}(a)))@>{%\rho}>{}>(K\Delta(a),V_{K\Delta(a)}(a))\end{CD}

appears. Here s is the natural regular map resolving singularities of KΔ(a), ρ and ρ are natural projections and R+s is a map R+ΔR+Δ(a) defined by s. The preimage of KΓ under ρ is the union of KΣ with ΣG(Δ) and CΔ(Σ)CΔ(a)(Γ). By 2.5.D  , the set of singular points of VKΔ(a) does not intersect KΣ, and VKΔ(a) is transversal to KΣ.

If ΣG(Δ), CΔ(Σ)CΔ(a)(Γ) and dimΣ=dimΓ, then R+s defines an isomorphism R+CΣ(Σ)R+CΓ(Γ), and if ΣG(Δ), CΔ(Σ)CΔ(a)(Γ) and dimΣ>dimΓ, then R+s defines a map R+CΣ(Σ)R+CΓ(Γ) which is a factorization by the action of quasi-homotheties qhw,t with wCΣ(Σ), wCΓ(Γ). By 2.5.E  , in the latter case variety VKΣ(aΓ) coinciding, by 2.5.A  , with VKΔ(a)KΣ is invariant under the same quasi-homotheties. Hence VKΔ(a)=s1VKΔ(a)(a) and hypersurface ρ1VKΔ(a), being the image of ρ1VKΔ(a) under R+×id, appears to be nonsingular along its intersection with R+Γ×UKn and transversal to R+Γ×UKn.

4. Patchworking

4.1. Patchworking L-polynomials

Let Δ, Δ1, …, ΔsRn be (convex integer) polyhedra with Δ=i=1sΔi and IntΔiIntΔj= for ij. Let ν:ΔR be a nonnegative convex function satisfying to the following conditions:

  1. (1)

    all the restrictions ν|Δi are linear;

  2. (2)

    if the restriction of ν to an open set is linear then this set is contained in one of Δi;

  3. (3)

    ν(ΔZn)Z.

 Remark 4.1.A  .

Existence of such a function ν is a restriction on a collection Δ1,,Δs. For example, the collection of convex polygons shown in Figure 27 does not admit such a function.

Figure 27.

Let a1,,as be Laurent polynomials over K in n variables with Δ(ai)=Δ. Let aiΔiΔj=ajΔiΔj for any i, j. Then, obviously, there exists an unique L-polynomial a with Δ(a)=Δ and aΔi=ai for i=1,,s. If a(x1,,xn)=ωZnaωxω, we put b(x,t)=ωZnaωxωtν(ω). This L-polynomial in n+1 variables is considered below also as a one-parameter family of L-polynomials in n variables. Therefore let me introduce the corresponding notation: put bt(x1,,xn)=b(x1,,xn,t). L-polynomials bt are said to be obtained by patchworking L-polynomials a1,,as by ν or, briefly, bt is a patchwork of L-polynomials a1,,as by ν.

4.2. Patchworking charts

Let a1,,as be Laurent polynomials over K in n variables with IntΔ(ai)IntΔ(aj)= for ij. A pair (Δ×UKn,υ) is said to be obtained by patchworking K-charts of Laurent polynomials a1,,as and it is a patchwork of K-charts of L-polynomials a1,,as if Δ=i=1sΔ(ai) and one can choose K-charts (Δ(ai)×UKn,υi) of Laurent polynomials a1,,as such that υ=i=1sυi.

4.3. The Main Patchwork Theorem

Let Δ, Δ1, …, Δs, ν, a1, …, as, b and bt be as in Section 4.1 (bt is a patchwork of L-polynomials a1, …, as by ν).

4.3.A  .

If L-polynomials a1,,as are completely nondegenerate then there exists t0>0 such that for any t(0,t0] a K-chart of L-polynomial bt is obtained by patchworking K-charts of L-polynomials a1,,as.

Proof.

Denote by G the union i=1sG(Δi). For ΓG denote by Γ~ the graph of ν|Γ. It is clear that Δ(b) is the convex hull of graph of ν, so Γ~G(Δ(b)) and thus there is an injection GG(Δ(b)):ΓΓ~. Restrictions Γ~Γ of the natural projection pr:Rn+1Rn are homeomorphisms, they are denoted by g.

Let p:Δ(b)×UKn+1KΔ(b) be the composition of the homeomorphism

\begin{CD}\Delta(b)\times U_{K}^{n+1}@>{h\times\operatorname{id}}>{}>{\mathbb{%R}}_{+}\Delta(b)\times U_{K}^{n+1}\end{CD}

and the natural projection ρ:R+Δ(b)×UKn+1KΔ(b) (cf. Section 3.3), so the pair (Δ(b)×UKn+1,p1VKΔ(b)(b)) is a K-chart of b. By 2.5.A  , for i=1,,s the pair

(Δ(ai)~×UKn+1,p1(VKΔ(b)(b)Δ(ai)~×UKn))

is a K-chart of L-polynomial bΔ(ai)~.

The pair

(Δ(ai)~×UKn,p1(VKΔ(b)(b)Δ(ai)~×UKn))

which is cut out by this pair on Δ(ai)~×UKn is transformed by g×id:Δ(ai)~×UKnΔ(ai)×UKn to a K-chart of ai. Indeed, g:Δ(ai)~Δ(ai) defines an isomorphism g:KΔ(ai)KΔ(ai)~ and since bΔ(ai)~(x1,,xn,1)=ai(x1,,xn), it follows that g:VKΔ(ai)(ai)=VKΔ(ai)~(bΔ(ai)~) and g defines a homeomorphism of the pair (Δ(ai)~×UKn,p1(VKΔ(b)(b)Δ(ai)~×UKn)) to a K-chart of L-polynomial ai.

Therefore the pair

(i=1sΔ(ai)~×UKn,p1(VKΔ(b)(b)i=1sΔ(ai)~×UKn))

is a result of patchworking K-charts of a1,,as.

For t>0 and ΓG(Δ) let us construct a ring homomorphism

K[CΔ(b)(Δ(b)pr1(Γ))Zn+1]K[CΔ(Γ)Zn]

which maps a monomial x1ω1xnωnxn+1ωn+1 to tωn+1x1ω1xnωn. This homomorphism corresponds to the embedding

KCΔ(Γ)KCΔ(b)(Δ(b)pr1(Γ))

extending the embedding

KRnKRn+1:(x1,,xn)(x1,,xn,t)

. The embeddings constructed in this way agree to each other and define an embedding KΔKΔ(b). Denote the latter embedding by it. It is clear that VKΔ(bt)=it1VKΔ(b)(b).

The sets ρ1itKΔ are smooth hypersurfaces of Δ(b)×UKn+1, comprising a smooth isotopy. When t0, the hypersurface ρ1itKΔ tends (in C1-sense) to

i=1sΔ(ai)~×UKn.

By 3.3.A  , ρ1VKΔ(b)(b) is transversal to each of

R+Δ(ai)~×UKn+1

and hence, the intersection ρ1(itKΔ)ρ1(VKΔ(b)(b)) for sufficiently small t is mapped to

VKΔ(b)(b)i=1sΔ(ai)~×UKn

by some homeomorphism

ρ1itKΔi=1sΔ(ai)~×UKn.

Thus the pair

(ρ1itKΔ,ρ1itKΔρ1VKΔ(b)(b))

is a result of patchworking K-charts of L-polynomials a1,,as if t belongs to a segment of the form (0,t0]. On the other hand, since VKΔ(bt)=it1VKΔ~(b),

ρ1itKΔρ1VKΔ(b)(b)=ρ1itVKΔ(bt)

and, hence, the pair

(ρ1itKΔ,ρ1itKΔρ1VKΔ(b)(b))

is homeomorphic to a K-chart of L-polynomial bt.

5. Perturbations smoothing a singularity of hypersurface

The construction of the previous Section can be interpreted as a purposeful smoothing of an algebraic hypersurface with singularities, which results in replacing of neighborhoods of singular points by new fragments of hypersurface, having a prescribed topological structure (cf. Section 1.10). According to well known theorems of theory of singularities, all theorems on singularities of algebraic hypersurfaces are extended to singularities of significantly wider class of hypersurfaces. In particular, the construction of perturbation based on patchworking is applicable in more general situation. For singularities of simplest types this construction together with some results of topology of algebraic curves allows to get a topological classification of perturbations which smooth singularities completely.

The aim if this Section is to adapt patchworking to needs of singularity theory.

5.1. Singularities of hypersurfaces

Let GKn be an open set, and let φ:GK be an analytic function. For UG denote by VU(φ) the set {xU|φ(x)=0}.

By singularity of a hypersurface VG(φ) at the point x0VG(φ) we mean the class of germs of hypersurfaces which are diffeomorphic to the germ of VG(φ) at x0. In other words, hypersurfaces VG(φ) and VH(ψ) have the same singularity at points x0 and y0, if there exist neighborhoods M and N of x0 and y0 such that the pairs (M,VM(φ)), (N,VN(ψ)) are diffeomorphic. When considering a singularity of hypersurface at a point x0, to simplify the formulas we shall assume that x0=0.

The multiplicity or the Milnor number of a hypersurface VG(φ) at 0 is the dimension

dimKK[[x1,,xn]]/(f/x1,,f/xn)

of the quotient of the formal power series ring by the ideal generated by partial derivatives f/x1,,f/xn of the Taylor series expansion f of the function φ at 0. This number is an invariant of the singularity (see [AVGZ82]). If it is finite, then we say that the singularity is of finite multiplicity.

If the singularity of VG(φ) at x0 is of finite multiplicity, then this singularity is isolated, i.e. there exists a neighborhood UKn of x0, which does not contain singular points of VG(φ). If K=C then the converse is true: each isolated singularity of a hypersurface is of finite multiplicity. In the case of isolated singularity, the boundary of a ball BKn, centered at x0 and small enough, intersects VG(φ) only at nonsingular points and only transversely, and the pair (B,VB(φ)) is homeomorphic to the cone over its boundary (B,VB(φ)) (see [Mil68], Theorem 2.10). In such a case the pair (B,VB(φ)) is called the link of singularity of VG(φ) at x0.

The following Theorem shows that the class of singularities of finite multiplicity of analytic hypersurfaces coincides with the class of singularities of finite multiplicity of algebraic hypersurfaces.

5.1.A  Tougeron’s theorem.

(see, for example, [AVGZ82], Section 6.3). If the singularity at x0 of a hypersurface VG(φ) has finite Milnor number μ, then there exist a neighborhood U of  x0 in Kn and a diffeomorphism h of this neighborhood onto a neighborhood of  x0 in Kn such that h(VU(φ))=Vh(U)(f(μ+1)), where f(μ+1) is the Taylor polynomial of φ of degree μ+1 .

The notion of Newton polyhedron is extended over in a natural way to power series. The Newton polyhedron Δ(f) of the series f(x)=ωZnaωxω (where xω=x1ω1x2ω2xnωn) is the convex hull of the set {ωRn|aω0}. (Contrary to the case of a polynomial, the Newton polyhedron Δ(f) of a power series may have infinitely many faces.)

However in the singularity theory the notion of Newton diagram occurred to be more important. The Newton diagram Γ(f) of a power series f is the union of the proper faces of the Newton polyhedron which face the origin, i.e. the union of the faces ΓG(Δ(f)) for which cones D+CΔ(f)(Γ) intersect the open positive orthant IntAn={xRn|x1>0,,xn>0}.

It follows from the definition of the Milnor number that, if the singularity of VG(φ) at 0 is of finite multiplicity, the Newton diagram of the Taylor series of φ is compact, and its distance from each of the coordinate axes is at most 1.

For a power series f(x)=ωZnfωxω and a set ΓRn the power series ωΓZnfωxω is called Γ-truncation of f and denoted by fΓ (cf. Section 2.1).

Let the Newton diagram of the Taylor series f of a function φ be compact. Then fΓ(f) is a polynomial. The pair (Γ(f)×UKn,γ) is said to be a nonreduced chart of germ of hypersurface VG(φ) at 0 if there exists a K-chart (Δ(fΓ(f)×UKn,υ) of fΓ(f) such that γ=υ(Γ(f)×UKn). It is clear that a nonreduced chart of germ of hypersurface is comprised of K-charts of fΓ, where Γ runs over the set of all faces of the Newton diagram.

A power series f is said to be nondegenerate if its Newton diagram is compact and the distance between it and each of the coordinate axes is at most 1 and for any its face Γ polynomial fΓ is completely nondegenerate. In this case about the germ of VG(φ) at zero we say that it is placed nondegenerately. It is not difficult to prove that nondegenerately placed germ defines a singularity of finite multiplicity. It is convenient to place the charts of germs of hypersurfaces in Kn by a natural map An×UKnKn:(x,y)Sy(x) (like K-charts of an L-polynomial, cf. Section 3.3). Denote by ΣK(φ) the image of Γ(f)×UKn under this map; the image of nonreduced chart of germ of hypersurface VG(φ) at zero under this map is called a (reduced) chart of germ of VG(φ) at the origin. It follows from Tougeron’s theorem that in this case adding a monomial of the form ximi to φ with mi large enough does not change the singularity. Thus, without changing the singularity, one can make the Newton diagram meeting the coordinate axes.

In the case when this takes place and the Taylor series of φ is nondegenerate there exists a ball UKn centered at 0 such that the pair (U,VU(φ)) is homeomorphic to the cone over a chart of germ of VG(φ). This follows from Theorem 5.1.A  and from results of Section 2.5.

Thus if the Newton diagram meets all coordinate axes and the Taylor series of φ is nondegenerate, then the chart of germ of VG(φ) at zero is homeomorphic to the link of the singularity.

5.2. Evolving of a singularity

Now let the function φ:GK be included as φ0 in a family of analytic functions φt:GK with t[0,t0]. Suppose that this is an analytic family in the sense that the function G×[0,t0]K:(x,t)φt(x) which is determined by it is real analytic. If the hypersurface VG(φ) has an isolated singularity at x0, and if there exists a neighborhood U of x0 such that the hypersurfaces VG(φt) with t[0,t0] have no singular points in U, then the family of functions φt with t[0,t0] is said to evolve the singularity of VG(φ) at x0.

If the family φt with t[0,t0] evolves the singularity of the hypersurface VG(φ0) at x0, then there exists a ball BKn centered at x0 such that

  1. (1)

    for t[0,t0] the sphere B intersects VG(φt) only in nonsingular points of the hypersurface and only transversely,

  2. (2)

    for t(0,t0] the ball B contains no singular point of the hypersurface VG(φt),

  3. (3)

    the pair (B,VB(φ0)) is homeomorphic to the cone over its boundary (B, VB(φ0)).

Then the family of pairs (B,VB(φt)) with t[0,t0] is called an evolving of the germ of VG(φ0) in x0. (Following the standard terminology of the singularity theory, it would be more correct to say not a on family of pairs, but rather a family of germs or even germs of a family; however, from the topological viewpoint, which is more natural in the context of the topology of real algebraic varieties, the distinction between a family of pairs satisfying 1 and 2 and the corresponding family of germs is of no importance, and so we shall ignore it.)

Conditions 1 and 2 imply existence of a smooth isotopy ht:BB with t(0,t0], such that ht0=id and ht(VB(φt0))=VB(φt), so that the pairs (B,VB(φt)) with t(0,t0] are homeomorphic to each other.

Given germs determining the same singularity, a evolving of one of them obviously corresponds to a diffeomorphic evolving of the other germ. Thus, one may speak not only of evolvings of germs, but also of evolvings of singularities of a hypersurface.

The following three topological classification questions on evolvings arise.

5.2.A  .

Up to homeomorphism, what manifolds can appear as VB(φt) in evolvings of a given singularity?

5.2.B  .

Up to homeomorphism, what pairs can appear as (B,VB(φt)) in evolvings of a given singularity?

Smoothings (B,VB(φt)) with t[0,t0] and (B,VB(φt)) with t[0,t0] are said to be topologically equivalent if there exists an isotopy ht:BB with t[0,min(t0,t0)], such that h0 is a diffeomorphism and VB(φt)=htVB(φt) for t[0,min(t0,t0)].

5.2.C  .

Up to topological equivalence, what are the evolvings of a given singularity?

Obviously, 5.2.B  is a refinement of 5.2.A  . In turn, 5.2.C  is more refined than 5.2.B  , since in 5.2.C  we are interested not only in the type of the pair obtained in result of the evolving, but also the manner in which the pair is attached to the link of the singularity.

In the case K=R these questions have been answered in literature only for several simplest singularities.

In the case K=C a evolving of a given singularity is unique from each of the three points of view, and there is an extensive literature (see, for example, [GZ77]) devoted to its topology (i.e., questions 5.2.A  and 5.2.B  ).

By the way, if we want to get questions for K=C which are truly analogous to questions 5.2.A  5.2.C  for K=R, then we have to replace evolvings by deformations with nonsingular fibers and one-dimensional complex bases, and the variety VB(φt) and the pairs (B,VB(φt)) have to be considered along with the monodromy transformations. It is reasonable to suppose that there are interesting connections between questions 5.2.A  5.2.C  for a real singularity and their counter-parts for the complexification of the singularity.

5.3. Charts of evolving

Let the Taylor series f of function φ:GK be nondegenerate and its Newton diagram meets all the coordinate axes. Let a family of functions φt:GK with t[0,t0] evolves the singularity of VG(φ) at 0. Let (B,VB(φt)) be the corresponding evolving of the germ of this hypersurface and ht:BB with t(0,t0] be an isotopy with ht0=id and ht(VB(φt0))=VB(φt)) existing by conditions 1 and 2 of the previous Section. Let (ΣK(φ),γ) be a chart of germ of hypersurface VG(φ) at zero and g:(ΣK(φ),γ)(B,VB(φ)) be the natural homeomorphism of it to link of the singularity.

Denote by ΠK(φ) a part of Kn bounded by ΣK(φ). It can be presented as a cone over ΣK(φ) with vertex at zero.

One can choose the isotopy ht:BB, t(0,t0] such that its restriction to B can be extended to an isotopy ht:BB with t[0,t0] (i.e., extended for t=0).

We shall call the pair (ΠK(φ), τ) a chart of evolving (B, VB(φt)), t[0,t0], if there exists a homeomorphism (ΠK(φ), τ)(B, VB(t0)), whose restriction ΣK(φ)B is the composition \begin{CD}\Sigma_{K}(\varphi)@>{g}>{}>\partial B@>{h^{\prime}_{0}}>{}>\partialB%\end{CD}. One can see that the boundary (ΠK(φ), τ) of a chart of evolving is a chart (ΣK(φ), γ) of the germ of the hypersurface at zero, and a chart of evolving is a pair obtained by evolving which is glued to (ΣK(φ), γ) in natural way. Thus that the chart of an evolving describes the evolving up to topological equivalence.

5.4. Construction of evolvings by patchworking

Let the Taylor series f of function φ:GK be nondegenerate and its Newton diagram Γ(f) meets all the coordinate axes.

Let a1,,as be completely nondegenerate polynomials over K in n variables with IntΔ(ai)IntΔ(aj)= and aiΔ(ai)Δ(aj)=ajΔ(ai)Δ(aj) for ij. Let i=1sΔ(ai) be the polyhedron bounded by the coordinate axes and Newton diagram Γ(f). Let aiΔ(ai)Δ(f)=fΔ(ai)Δ(f) for i=1,,s. Let ν:i=1sΔ(ai)R be a nonnegative convex function which is equal to zero on Γ(f) and satisfies conditions 1, 2, 3 of Section 4.1 with polyhedra Δ(a1), …, Δ(as). Then polynomials a1,,as can be ”glued to φ by ν” in the following way generalizing patchworking L-polynomials of Section 4.1.

Denote by a the polynomial defined by conditions aΔ(ai)=ai for i=1,,s and ai=1sΔ(ai)=a. If a(x)=ωZnaωxω then we put

φt(x)=φ(x)+(ωZnaωxωtν(ω))aΓ(f)x.
5.4.A  .

Under the conditions above there exists t0>0 such that the family of functions φt:GK with t[0,t0] evolves the singularity of VG(φ) at zero. The chart of this evolving is patchworked from K-charts of a1,,as.

In the case when φ is a polynomial, Theorem 5.4.A  is a slight modification of a special case of Theorem 4.3.A  . Proof of 4.3.A  is easy to transform to the proof of this version of 5.4.A  . The general case can be reduced to it by Tougeron Theorem, or one can prove it directly, following to scheme of proof of Theorem 4.3.A  .

We shall call the evolvings obtained by the scheme described in this Section patchwork evolvings.

6. Approximation of hypersurfaces of KRn

6.1. Sufficient truncations

Let M be a smooth submanifold of a smooth manifold X. Remind that by a tubular neighborhood of M in X one calls a submanifold N of X with MIntN equipped with a tubular fibration, which is a smooth retraction p:NM such that for any point xM the preimage p1(x) is a smooth submanifold of X diffeomorphic to DdimXdimM. If X is equipped with a metric and each fiber of the tubular fibration p:NM is contained in a ball of radius ε centered in the point of intersection of the fiber with M, then N is called a tubular ε-neighborhood of M in X.

We need tubular neighborhoods mainly for formalizing a notion of approximation of a submanifold by a submanifold. A manifold presented as the image of a smooth section of the tubular fibration of a tubular ε-neighborhood of M can be considered as sufficiently close to M: it is naturally isotopic to M by an isotopy moving each point at most by ε.

We shall consider the space Rn×UKn as a flat Riemannian manifold with metric defined by the standard Euclidian metric of Rn in the case of K=R and by the standard Euclidian metric of Rn and the standard flat metric of the torus UCn=(S1)n in the case of K=C.

An ε-sufficiency of truncations of Laurent polynomial defined below and the whole theory related with this notion presuppose that it has been chosen a class of tubular neighborhoods of smooth submanifolds of Rn×UKn invariant under translations Tω×idUKn and that for any two tubular neighborhoods N and N of the same M, which belong to this class, restrictions of tubular fibrations p:NM and p:NM to NN coincide. One of such classes is the collection of all normal tubular neighborhoods, i.e. tubular neighborhoods with fibers consisting of segments of geodesics which start from the same point of the submanifold in directions orthogonal to the submanifold. Another class, to which we shall turn in Sections 6.7 and 6.8, is the class of tubular neighborhoods whose fibers lie in fibers Rn1×t×UKn1×s of Rn×UKn and consist of segments of geodesics which are orthogonal to intersections of the corresponding manifolds with these Rn1×t×UKn1×s. The intersection of such a tubular neighborhood of M with the fiber Rn1×t×UKn1×s is a normal tubular neighborhood of M(Rn1×t×UKn1×s) in Rn1×t×UKn1×s. Of course, only manifolds transversal to Rn1×t×UKn1×s have tubular neighborhoods of this type.

Introduce a norm in vector space of Laurent polynomials over K on n variables:

||ωZnaωxω||=max{|aω||ωZn}.

Let Γ be a subset of Rn and ε a positive number. Let a be a Laurent polynomial over K in n variables and U a subset of KRn. We shall say that in U the truncation aΓ is ε-sufficient for a (with respect to the chosen class of tubular neighborhoods), if for any Laurent polynomial b over K satisfying the conditions Δ(b)Δ(a), bΓ=aΓ and ||bbΓ||||aaΓ|| (in particular, for b=a and b=aΓ) the following condition takes place:

  1. (1)

    USVKRn(b)=,

  2. (2)

    the set la(UVKRn(b)) lies in a tubular ε-neighborhood N (from the chosen class) of la(VKRn(aΓ)SVKRn(aΓ)) and

  3. (3)

    la(UVKRn(b)) can be extended to the image of a smooth section of the tubular fibration Nla(VKRn(aΓ)SVKRn(aΓ)).

The ε-sufficiency of Γ-truncation of Laurent polynomial a in U means, roughly speaking, that monomials which are not in aΓ have a small influence on VKRn(a)U.

6.1.A  .

If aΓ is ε-sufficient for a in open sets Ui with iJ, then it is ε-sufficient for a in iJUi too.

Standard arguments based on Implicit Function Theorem give the following Theorem.

6.1.B  .

If a set UKRn is compact and contains no singular points of a hypersurface VKRn(a), then for any tubular neighborhood N of VKRn(a)SVKRn(a) and any polyhedron ΔΔ(a) there exists δ>0 such that for any Laurent polynomial b with Δ(b)Δ and ||ba||<δ the hypersurface VKRn(b) has no singularities in U, intersection UVKRn(b) is contained in N and can be extended to the image of a smooth section of a tubular fibration NVKRn(a)SVKRn(a).

From this the following proposition follows easily.

6.1.C  .

If UKRn is compact and aΓ is ε-sufficient truncation of a in U, then for any polyhedron ΔΔ(a) there exists δ>0 such that for any Laurent polynomial b with Δ(b)Δ, bΓ=aΓ and ||ba||<δ the truncation bΓ is ε-sufficient in U.

In the case of Γ=Δ(a) proposition 6.1.C  turns to the following proposition.

6.1.D  .

If a set UKRn is compact and contains no singular points of VKRn(a) and la(VKRn(a)) has a tubular neighborhood of the chosen type, then for any ε>0 and any polyhedron ΔΔ(a) there exists δ>0 such that for any Laurent polynomial b with Δ(b)Δ, ||ba||<δ and bΔ(a)=a the truncation bΔ(a) is ε-sufficient in U.

The following proposition describes behavior of the ε-sufficiency under quasi-homotheties.

6.1.E  .

Let a be a Laurent polynomial over K in n variables. Let UKRn, ΓRn, wRn. Let ε and t be positive numbers. Then ε-sufficiency of Γ-truncation aΓ of a in qhw,t(U) is equivalent to ε-sufficiency of Γ-truncation of aqhw,t in U.

The proof follows from comparison of the definition of ε-sufficiency and the following two facts. First, it is obvious that

qhw,t(U)VKRn(b)=qhw,t(Uqhw,t1(VKRn(b)))=qhw,t(UVKRn(bqhw,t)),

and second, the transformation T(lnt)w×idUKn of Rn×UKn corresponding, by 2.1.A  , to qhw,t preserves the chosen class of tubular εneighborhoods .

6.2. Domains of ε-sufficiency of face-truncation

For ARn and BKRn denote by qhA(B) the union ωAqhω(B).

For ARn and ρ>0 denote by Nρ(A) the set {xRn|dist(x,A)<ρ}.

For A,BRn and λR the sets {x+y|xA,yB} and {λx|xA} are denoted, as usually, by A+B and λA.

Let a be a Laurent polynomial in n variables, ε a positive number and Γ a face of the Newton polyhedron Δ=Δ(a).

6.2.A  .

If in open set UKRn the truncation aΓ is ε-sufficient for a, then it is ε-sufficient for a in qhClDCΔ(Γ)(U).55Here (as above) Cl denotes the closure.

Proof.

Let ωClDCΔ(Γ) and ωw=δ for wΓ. By 6.1.E  , ε-sufficiency of truncation aΓ for a in qhω(U) is equivalent to ε-sufficiency of truncation (aqhω)Γ for (aqhω)Γ in U or, equivalently, to ε-sufficiency of Γ-truncation of Laurent polynomial b=eδaqhω in U. Since

eδaqhω(x)=wΔeδawxweωw=aΓ(x)+wΔΓeωwδawxw

and ωwδ0 when wΔΓ and ωClDCΔ(Γ), it follows that b satisfies the conditions Δ(b)=Δ, bΓ=aΓ and ||bbΓ||||aaΓ||. Therefore the truncation bΓ is ε-sufficient for b in U and, hence, the truncation aΓ is ε-sufficient for a in qhω(U). From this, by 6.1.A  , the proposition follows.

6.2.B  .

If the truncation aΓ is completely nondegenerate and laVKRn(aΓ) has a tubular neighborhood of the chosen type, then for any compact sets CKRn and ΩDCΔ(Γ) there exists δ such that in qhδΩ(C) the truncation aΓ is ε-sufficient for a.

Proof.

For ωDCΔ(Γ) denote by ωΓ a value taken by the scalar product ωw for wΓ. Since

tωΓaqhω,t(x)=aΓ(x)+wΔΓtωwωΓawxw

for ωDCΔ(Γ) (cf. the previous proof) and ωwωΓ<0 when wΔΓ and ωDCΔ(Γ) it follows that the Laurent polynomial bω,t=tωΓaqhω,t with ωDCΔ(Γ) turns to aΓ as Γ+. It is clear that this convergence is uniform with respect to ω on a compact set ΩDCΔ(Γ). By 6.1.D  it follows from this that for a compact set UKRn there exists η such that for any ωΩ and tη the truncation bω,tΓ of bω,t is ε-sufficient in U for bω,t. By 6.1.E  , the latter is equivalent to ε-sufficiency of truncation aΓ for a in qhω,t(U).

Thus if U is the closure of a bounded neighborhood W of a set C then there exists η such that for ωΩ and tη the truncation aΓ is ε-sufficient for a in qhω,t(U). Therefore aΓ is the same in a smaller set qhω,t(W) and, hence, (by 6.1.A  ) in the union tη,ωΩqhω,t(W) and, hence, in a smaller set t=η,ωΩ(C). Putting δ=lnη we obtain the required result.

6.2.C  .

Let Γ is a face of another face Σ of the polyhedron Δ. Let Ω is a compact subset of the cone DCΔ(Σ). If Γ-truncation aΓ is ε-sufficient for aΣ in a compact set C, then there exists a number δ such that aΓ is ε-sufficient for a in qhδΩ(C).

This proposition is proved similarly to 6.2.B  , but with the following difference: the reference to Theorem 6.1.D  is replaced by a reference to Theorem 6.1.C  .

6.2.D  .

Let CKRn be a compact set and let Γ be a face of Δ such that for any face Σ of Δ with dimΣ=dimΔ1 having a face Γ the truncation aΓ is ε-sufficient for aΣ in C. Then there exists a real number δ such that the truncation aΓ is ε-sufficient for a in qhClDCΔ(Γ)NδDCΔ(Δ)(IntC).

Proof.

By 6.2.C  , for any face Σ of Δ with dimΣ=dimΔ1 and ΓΣ there exists a vector ωΣDCΔ(Σ) such that the truncation aΓ is ε-sufficient for a in qhωΣ(C), and, hence, by 6.2.A  , in qhωΣ+ClDCΔ(Γ)(IntC). Choose such ωΣ for each Σ with dimΣ=dimΔ1 and ΓΣ. Obviously, the sets ωΣ+ClDCΔ(Γ) cover the whole closure of the cone DCΔ(Γ) besides some neighborhood of its top, i.e. the cone DCΔ(Δ); in other words, there exists a number δ such that Σ(ωΣ+ClDCΔ(Γ))ClDCΔ(Γ)NδDCΔ(Δ). Hence, aΓ is ε-sufficient for a in qhClDCΔ(Γ)NδDCΔ(Δ)(IntC)ΣqhωΣ+ClDCΔ(Γ)(IntC).

6.3. The main Theorem on logarithmic asymptotes of hypersurface

Let ΔRn be a convex closed polyhedron and φ:G(Δ)R be a positive function. Then for ΓG(Δ) denote by DΔ,φ(Γ) the set

Nφ(Γ)(DCΔ(Γ))ΣG(Δ), ΓG(Σ)Nφ(Σ)(DCΔ(Σ)).

It is clear that the sets DΔ,φ(Γ) with ΓG(Δ) cover Rn. Among these sets only sets corresponding to faces of the same dimension can intersect each other. In some cases (for example, if φΓ grows fast enough when dimΓ grows) they do not intersect and then {DΔ,φ(Γ)}ΓG(Δ) is a partition of Rn.

Let a be a Laurent polynomial over K in n variables and ε be a positive number. A function φ:G(Δ(a))Rn is said to be describing domains of ε-sufficiency for a (with respect to the chosen class of tubular neighborhoods) if for any proper face ΓG(Δ(a)), for which truncation aΓ is completely non-degenerate and the hypersurface la(VKRn(aΓ)) has a tubular neighborhood of the chosen class, the truncation aΓ is ε-sufficient for a in some neighborhood of l1(DCΔ(a),φ(Γ)).

6.3.A  .

For any Laurent polynomial a over K in n variables and ε>0 there exists a function G(Δ(a))R describing domains of ε-sufficiency for a with respect to the chosen class of tubular neighborhoods.

In particular, if a is peripherally nondegenerate Laurent polynomial over K in n variables and dimΔ(a)=n then for any ε>0 there exists a compact set CKRn such that KRnC is covered by regions in which truncations of aΔ(a) are ε-sufficient for a with respect to class of normal tubular neighborhoods. In other words, under these conditions behavior of VKRn(a) outside C is defined by monomials of aΔ(a).

6.4. Proof of Theorem 6.3.A 

Theorem 6.3.A  is proved by induction on dimension of polyhedron Δ(a).

If dimΔ(a)=0 then a is monomial and VKRn(a)=. Thus for any ε>0 any function φ:G(Δ(a))R describes domains of ε-sufficiency for a.

Induction step follows obviously from the following Theorem.

6.4.A  .

Let a be a Laurent polynomial over K in n variables, Δ be its Newton polyhedron, ε a positive number. If for a function φ:G(Δ){Δ}R and any proper face Γ of Δ the restriction φ|G(Γ) describes domains of ε-sufficiency for aΓ, then φ can be extended to a function φ¯:G(Δ)R describing regions of ε-sufficiency for a.

Proof.

It is sufficient to prove that for any face ΓG(Δ){Δ}, for which the truncation aΓ is completely nondegenerate and hypersurface VKRn(aΓ) has a tubular neighborhood of the chosen class, there exists an extension φΓ of φ such that truncation aΓ is ε-sufficient for a in a neighborhood of l1(DΔ,φΓ(Γ)), i.e. to prove that for any face ΓΔ there exists a number φΓ(Δ) such that the truncation aΓ is ε-sufficient for a in some neighborhood of

l1(Nφ(Γ)(DCΔ(Γ))[NφΓ(Δ)(DCΔ(Δ))ΣG(Δ){Δ}, ΓG(Σ)Nφ(Σ)(DCΔ(Σ))].

Indeed, putting

φ¯(Δ)=maxΓG(Δ){Δ}φΓ(Δ)

we obtain a required extension of φ.

First, consider the case of a face Γ with dimΓ=dimΔ1. Apply proposition 6.2.B  to C=l1(ClNφ(Γ)+1(0) and any one-point set ΩDCΔ(Γ). It implies that aΓ is ε-sufficient for a in qhω(C)=l1(ClNφ(Γ)+1(ω)) for some ωDCΔ(Γ). Now apply proposition 6.2.A  to U=l1(Nφ(Γ)+1(ω)). It gives that aΓ is ε-sufficient for a in qhDCΔ(Γ)(l1(Nφ(Γ)+1(ω))=l1(Nφ(Γ)+1(ω+DCΔ(Γ))) and, hence, in the smaller set l1(Nφ(Γ)+1(DCΔ(Γ)))N|ω|(DCΔ(Δ)). It is remained to put φΓ(Δ)=|ω|+1.

Now consider the case of face Γ with dimΓ<dimΔ1. Denote by E the set

Nφ(Γ)(DCΔ(Γ))ΣG(Δ){Δ}, ΓG(Σ)Nφ(Σ)(DCΔ(Σ)).

It is clear that there exists a ball BRn with center at 0 such that E=(EB)+ClDCΔ(Γ). Denote the radius of this ball by β.

If ΣG(Δ) is a face of dimension dimΔ1 with ΣΓ then, by the hypothesis, the truncation aΓ is ε-sufficient for aΣ in some neighborhood of

l^{-1}(\mathfrak{N}_{\varphi(\Gamma)}(DC^{-}_{\Sigma}(\Gamma))\smallsetminus%\bigcup_{\Theta\in\mathcal{G}(\Sigma),\ \Gamma\in\mathcal{G}(\Theta)}\mathfrak%{N}_{\varphi(\Theta)}(DC^{-}_{\Sigma}(\Theta))

and, hence, in neighborhood of a smaller set

l^{-1}(\mathfrak{N}_{\varphi(\Gamma)}(DC^{-}_{\Delta}(\Gamma))\smallsetminus%\bigcup_{\Theta\in\mathcal{G}(\Sigma),\ \Gamma\in\mathcal{G}(\Theta)}\mathfrak%{N}_{\varphi(\Theta)}(DC^{-}_{\Delta}(\Theta)).

Therefore for any face Σ with dimΣ=dimΔ1 and ΓΣ the truncation aΓ is ε-sufficient for aΣ in some neighborhood of l1(E). Denote by C a compact neighborhood of l1(EB) contained in this neighborhood. Applying proposition 6.2.A  , one obtains that aΓ is ε-sufficient for a in the set

qhClDCΔ(Γ)NδDCΔ(Δ)(IntC)=l1(Intl(C)+ClDCΔ(Γ)NδDCΔ(Δ))).

It is remained to put φΓ(Δ)=δ+β

6.5. Modification of Theorem 6.3.A 

Below in Section 6.8 it will be more convenient to use not Theorem 6.3.A  but the following its modification, whose formulation is more cumbrous, and whose proof is obtained by an obvious modification of deduction of 6.3.A  from 6.4.A  .

6.5.A  .

For any Laurent polynomial a over K in n variables and any ε>0 and c>1 there exists a function φ:G(Δ(a))R such that for any proper face ΓG(Δ(a)), for which aΓ is completely nondegenerate and la(VKRn(aΓ)) has a tubular neighborhood from the chosen class, the truncation aΓ is ε-sufficient for a in some neighborhood of

l^{-1}(\mathfrak{N}_{c\varphi(\Gamma)}(DC^{-}_{\Delta}(\Gamma))\smallsetminus%\bigcup_{\Sigma\in\mathcal{G}(\Delta),\ \Gamma\in\mathcal{G}(\Sigma)}\mathfrak%{N}_{\varphi(\Sigma)}(DC^{-}_{\Delta}(\Sigma)).

6.6. Charts of L-polynomials

Let a be a peripherally nondegenerate Laurent polynomial over K in n variables, Δ be its Newton polyhedron. Let V be a vector subspace of Rn corresponding to the smallest affine subspace containing Δ (i.e. V=CΔ(Δ)). Let φ:G(Δ)R be the function, existing by 6.3.A  , describing for some ε regions of ε-sufficiency for a with respect to class of normal tubular neighborhoods.

The pair (Δ×UKn, υ) consisting of the product Δ×UKn and its subset υ is a K-chart of a Laurent polynomial a if:

  1. (1)

    there exists a homeomorphism h:(ClDΔ,φ(Δ)V)×UKnΔ×UKn such that h((ClaDΔ,φ(Δ)V)×y)=Δ×y for yUKn,

    υ=h(laVKRn(a)(ClDΔ,φ(Δ)V)×UKn

    and for each face Γ of Δ the set h((ClDΔ,φ(Δ)DΔ,φ(Γ)V)×UKn) lies in the product of the star ΓG(Σ)ΣG(Δ){Δ}Σ of Γ to UKn;

  2. (2)

    for any vector ωRn, which is orthogonal to V and, in the case of K=R, is integer, the set υ is invariant under transformation Δ×UKnΔ×UKn defined by formula (x, (y1,…, yn))(x, (eπiω1y1, …, eπiωnyn));

  3. (3)

    for each face Γ of Δ the pair (Γ×UKn, υ(Γ×UKn)) is a K-chart of Laurent polynomial aΓ.

The definition of the chart of a Laurent polynomial, which, as I believe, is clearer than the description given here, but based on the notion of toric completion of KRn, is given above in Section 3.3. I restrict myself to the following commentary of conditions 13.

The set (ClDΔ,φ(Δ)V)×UKn contains, by 6.3.A  , a deformation retract of laVKRn(a). Thus, condition 1 means that υ is homeomorphic to a deformation retract of VKRn(a). The position of υ in Δ×UKn contains, by 1 and 3, a complete topological information about behavior of this hypersurface outside some compact set. The meaning of 2 is in that υ has the same symmetries as, according to 2.1.C  , VKRn(a) has.

6.7. Structure of VKRn(bt) with small t

Denote by it the embedding KRnKRn+1 defined by it(x1,,xn)=(x1,,xn,t). Obviously,

VKRn(bt)=it1VKRn+1(b).

This allows to take advantage of results of the previous Section for study of VKRn(bt) as t0. For sufficiently small t the image of embedding it is covered by regions of ε-sufficiency of truncation bΓ~, where Γ~ runs over the set of faces of graph of ν, and therefore the hypersurface VKRn(bt) turns to be composed of pieces obtained from VKRn(ai) by appropriate quasi-homotheties.

I preface the formulation describing in detail the behavior of VKRn(bt) with several notations.

Denote the Newton polyhedron Δ(b) of Laurent polynomial b by Δ~. It is clear that Δ~ is the convex hull of the graph of ν. Denote by G the union i=1sG(Δi). For ΓG denote by Γ~ the graph ν|Γ. It is clear that Γ~G(Δ~) and hence an injection ΓΓ~:GG(Δ~) is defined.

For t>0 denote by jt the embedding RnRn+1 defined by the formula jt(x1,,xn)=(x1,,xn,lnt). Let ψ:GR be a positive function, t be a number from interval (0,1). For ΓG denote by Et,ψ(Γ) the following subset of Rn:

Nψ(Γ)jt1(DCΔ~(Γ~))ΣG, ΓG(Σ)Nφ(Σ)jt1(DCΔ~(Σ~)).
6.7.A  .

If Laurent polynomials a1,,as are completely non-degenerate then for any ε>0 there exist t0(0,1) and function ψ:GR such that for any t(0,t0] and any face ΓG truncation btΓ is ε-sufficient for bt with respect to the class of normal tubular neighborhoods in some neighborhood of l1(Et,ψ(Γ)).

Denote the gradient of restriction of ν on ΓG by (Γ). The truncation btΓ, obviously, equals aΓqh(Γ),t. In particular, btΔi=aiqh(Δi),t and, hence,

VKRn(btΔi)=qh(Δi),t1(VKRn(ai)).

In the domain, where btΓ is ε-sufficient for bt, the hypersurfaces laVKRn(bt) and laVKRn(btΔi) with ΔiΓ lie in the same normal tubular ε-neighborhood of laVKRn(btΓ) and, hence, are isotopic by an isotopy moving points at most on 2ε. Thus, according to 6.7.A  , for tt0 to the space KRn is covered by regions in which VKRn(bt) is approximated by qh(Δi),t1(VKRn(ai)).

6.8. Proof of Theorem 6.7.A 

Put c=max{1+(Δi)2, i=1,,s}. Apply Theorem 6.5.A  to the Laurent polynomial b and numbers ε and c, considering as the class of chosen tubular neighborhoods in Rn+1×UKn+1 tubular neighborhoods, whose fibers lie in the fibers Rn×t×UKn×s of Rn+1×UKn+1 and consist of segments of geodesics which are orthogonal to intersections of submanifold with Rn×t×UKn×s. (Intersection of such a tubular neighborhood of MRn+1×UKn+1 with the fiber Rn×t×UKn×s is a normal tubular neighborhood of M(Rn×t×UKn×s) in Rn×t×UKn×s.) Applying Theorem 6.5.A  one obtains a function φ:G(Δ~)R. Denote by ψ the function GR which is the composition of embedding ΓΓ~:GG(Δ~) (see Section 6.7) and the function 1cφ:G(Δ~)R. This function has the required property. Indeed, as it is easy to see, for 0<t<eφ(Δ~) Et,ψ(Γ) is contained, in

jt1(Ncφ(Γ~)(DCΔ~(Γ~))Σ~G(Δ~), Γ~G(Σ~)Nφ(Σ~)(DCΔ~(Σ~))),

and thus from ε-sufficiency of bΓ~ for b in some neighborhood of

l1(Ncφ(Γ~)(DCΔ~(Γ~))Σ~G(Δ~), Γ~G(Σ~)Nφ(Σ~)(DCΔ~(Σ~))),

with respect to the chosen class of tubular neighborhoods in Rn+1×UKn+1 if follows that for 0<t<eφ(Δ~) the truncation btΓ is ε-sufficient for bt in some neighborhood of l1(Et,ψ(Γ)) with respect to the class of normal tubular neighborhoods.

6.9. An alternative proof of Theorem 4.3.A 

Let V be a vector subspace of Rn corresponding to the minimal affine subspace containing Δ. It is divided for each t(0,1) onto the sets Vjt1(DCΔ~(Γ~)) with ΓG. Let us construct cells Γt in V which are dual to the sets of this partition (barycentric stars). For this mark a point in each Vjt1(DCΔ~(Γ~)):

bt,ΓVjt1(DCΔ~(Γ~)).

Then for Γ with dimΓ=0 put Γt=bt,Γ and construct the others Γt inductively on dimension dimΓ: if Γt for Γ with dimΓ<r have been constructed then for Γ with dimΓ=r the cell Γt is the (open) cone on ΣG(Γ){Γ}Σt with the vertex bt,Γ. (This is the usual construction of dual partition turning in the case of triangulation to partition onto barycentric stars of simplices.)

By Theorem 6.7.A  there exist t0(0,1) and function ψ:GR such that for any t(0,t0] and any face ΓG the truncation btΓ is ε-sufficient for bt in some neighborhood of l1(Et,ψ(Γ)). Since cells Γt grow unboundedly when t runs to zero (if dimΓ0) it follows that there exists t0(0,t0] such that for t(0,t0] for each face ΓG the set Nψ(Γ)jt1(DCΔ~(Γ)), and together with it the set Et,ψ(Γ), lie in the star of the cell Γt, i.e. in ΓG(Σ)Σt. Let us show that for such t0 the conclusion of Theorem 4.3.A  takes place.

Indeed, it follows from 6.7.A  that there exists a homeomorphism h:Γt×UKnΓ×UKn with h(Γt×y)=Γ×y for yUKn such that (Γ×UKn, h(la(VKRn(bt))Γt×UKn)) is K-chart of Laurent polynomial aΓ. Therefore the pair

(ΓGΓt×UKn,laVKRn(bt)(ΓGΓt×UKn))

is obtained in result of patchworking K-charts of Laurent polynomials a1, …, as. The function φ:G(Δ)R, existing by Theorem 6.3.A  applied to bt, can be chosen, as it follows from 6.4.A  , in such a way that it should majorate any given in advance function G(Δ)R. Choose φ in such a way that Dφ,Δ(Δ)ΓGΓt and Dφ,Δ(Σ)Dφ,Δ(Δ)Et,ψ(Σ)Dφ,Δ(Δ) for ΣG(Δ){Δ}. As it follows from 6.7.A  , there exists a homeomorphism

(1) (ΓGΓt×UKn,laVKRn(bt)(ΓGΓt×UKn))(Dφ,Δ(Δ)×UKn,laVKRn(bt)(Dφ,Δ(Δ)×UKn))

turning Et,ψ(Σ)(ΓGΓt×UKn) to Et,ψ(Σ)Dφ,Δ(Δ) for ΣG(Δ){Δ}. Therefore K-chart of Laurent polynomial bt is obtained by patchworking K-charts of Laurent polynomials a1,,as.

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